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The initial plan for Fall Blau was to surround and destroy Soviet armies in the southern sector of the front, just like they did the previous year, and only then to move towards their main objective (i.e. Caucasus oil) and secondary objectives (Stalingrad, Black Sea ports, possibly Astrakhan etc ..). Due to reasons that are beyond the scope of this answer, this did not happen. The Soviets began a disorganized retreat, with huge losses, but were not completely destroyed.
The situation in August/September of 1942 begun to get complicated for the Germans as Soviet resistance stiffened, and their own forces were getting stretched and away from their supply bases. On the coast, the Germans and Romanians were having a problem capturing Novorossiysk. They would eventually succeed on 11th of September, but could not go much farther (the front would remain static there). In the direction of Stalingrad, after the battle of Kalach, fighting now moved into city itself. This is a well-known topic, so we won't dwell on it; it is sufficient to say that this part of the front also became mostly static in September.
This brings us to a main point, i.e. the drive towards oil fields (Grozny), and the simultaneous attempt to cross the Caucasus mountains and cut-off Soviet forces on the coast (thus securing their right flank). Soviet historiography considers the latter to have begun on the 9th of August 1942 with the fall of the town of Aramvir. At that moment, the Germans were breaking through into the northern Caucasus plains, and there was a danger they would turn southwards through the mountains towards the port of Tuapse. The Tuapse defensive operation, fought by elements of Transcaucasian front vs elements of Army Group A, is little known to the Western public. Without going into too many details, the German advance in this sector was slow and bloody (as would be expected in mountain warfare) before being completely stopped in November.
Further south-east, the Germans attempted to reach the Black Sea coast at Sukhumi, and this is the place where the famous capture of Mount Elbrus happened. It should be noted at this point Soviet forces were weaker than the were at Tuapse, simply because they did not expect a German offensive in this direction (worse terrain). In this part of the front (High Caucasus) snow begins sometimes even at the end of August, roads are barely mud tracks, so the whole expedition was of little military value. In any case, the Soviets managed to block this direction with little effort, since the Germans could not push more then few battalions of mountain troops in that direction.
Near Grozny, Soviets built their defenses on river Terek as a natural obstacle to the German drive towards Chechnya. The Germans crossed the river near Mozdok, thus beginning a battle known as the Mozdok-Maglobeg defensive operation. In the course of the battle that lasted whole of September, Soviets tried to reduce thr German bridgehead, while the Germans were attempting to expand it. The Germans moved their 5th SS Wiking division from around Tuapse, and finally managed to capture town of Maglobek which is about 30-40 km from the river, but at that point their efforts were being contained.
The final German effort came in late October and early November of 1942, at first having surprisingly good success due to unexpected direction (from Nalchik towards Vladikavkaz, then Ordzhonikidze), and became known as Nalchik-Ordzhonikidze Defensive Operation. Surprise came mostly because the Germans secretly regrouped 1st Panzer Army. However, by this point, the Soviets had enough reserves available to contain this offensive (in fact being readied for the Soviets' own offensive) before the Germans could achieve their strategic goals. It should be noted that in this particular operation, VVS actually started to appear over the battlefield, and managed to successfully contest Luftwaffe air superiority for practically the first time since the German offensive in the South started. Also, Soviet anti-tank capabilities (anti-tank guns above everything else) were increased, and in some sectors the Germans simply could not push trough with their panzers like they used to.
Overall, as the Germans expanded from the initial starting point of Fall Blau deep into Soviet territory, their troop concentration became diminished, especially after sustaining losses and moving away from their logistic points. Mountainous and overall bad terrain certainly did slow them down. Despite occasional successes of German or even Romanian mountain troops, they were not able to maneuver large formations and win decisive battles in such terrain. Even in relatively flatter terrain, obstacles like rivers slowed and sometimes completely stopped them. It should be noted that bad weather certainly played its part, as the Luftwaffe was unable to give full support (unlike for example in the summer months). Thus, blitzkrieg gradually gave way to a war of attrition which was ultimately more favorable to the Soviets.
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TL;DR: Terrain does not win battles, soldiers win battles. Your question is explained by the chosen german objectives and where soviet decided to defend those objectives.
In a operational area so huge as the Caucasus is, one first has to take into account the following:
Hence. War in such places is concentrated around (decent) roads. Think in the war in north Africa, which was carried primary near the coast, not deep in the desert. Any place far away of roads is not suitable for war. The capture of Mt. Elbrus was only to add a line in a sports almanac, but actually useless for military purposes because is too far away from roads for supply.
What happened in the Caucasus front is similar to what happened in Stalingrad. In Stalingrad, soviets did not defend west bank of Don river (when they did it, it was because they thought the germans where weak, but they were not), they started to give battle in the east bank of the river... they let germans advance giving little battle. Real battle was given in the Volga, in Stalingrad. Now, in the Caucasus is the same issue. Why defend useless desert, where your forces are going to be easily encircled, while having your supply depots really far away once Rostov falls? That's why germans advanced so easily in the north Caucasus, soviets let them do it.
Now, taking that into account, to understand why germans stalled their advance, you must see the battle from both points of views:
German point of view: First, the list of objectives.
Flat lands in the north-east of that list of places were not german objectives. That's why you don't see battles there.
The soviet point of view:
Taking the list of german objectives, next question is where is more convinient to defend? Mountain roads, coastal roads and rivers. Russians gave battle in those places (the coast, mountain passes and Terek river), and there they stopped germans. Maybe we should give merit to soviets, not only blame the geography.
In the chosen places soviets built fortifications, according to general Tiulenev:
In few weeks the whole Caucasus theater of war was turned into a net of fortifications. People worked until fall exhausted, covering their hands full of blisters with rags full of blood. Sometimes they had nothing for eating for days, but they still worked even during nights. By early autumn they had builded 100.000 defensive positions, including 70.000 casamates and forts... More than 800 kilometers of antitank ditches, 300 kilometers of obstacles for infantry, 1500 kilometers of trenchs, etc. In total, 9.150.000 workdays.
Germans stalled almost immediately when they reached soviet defenses. Same thing happened in Kursk later, germans attacked prepared defenses and they stalled quite fast.
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The heavily mechanized German army in the Caucasus ran itself ragged. Every advance took it further away from its supplies, further away from its air support, and wore their offensive elements down further. The Soviets retreated in decent order before them keeping up resistance until the Germans could not gather the resources to both hold what they had and advance further.
Then the western wing advanced deep in the Caucasus Mountains, capturing Mt. Elbrus, the highest elevation, about two weeks later. In theory, it was "downhill" from there. Yet the German drive stalled after, rather than before, reaching this summit.
In a video game, sticking a flag on the highest peak might end the battle for a mountain range. In reality, it's just for propaganda.
The Germans never made it out of the Greater Caucasus. Along most of the front they never made it into the Greater Caucasus. Nalchik, for example, is only at the foothills.
The Greater Caucasus are a broad, high, heavily forested mountain range spanning the entire 1200 km width from sea to sea. Reaching the highest peak doesn't mean it's all downhill from there, there's many peaks over 4000m. The fighting would have to continue through very rough terrain with poor roads and many, many defensible positions. If the front line had pushed south of the Greater Caucusus, they Germans would still have to hump all their supplies over that mountain range with its poor roads and deteriorating weather.
Towards the end of the campaign the Germans finally captured a major port on the Black Sea, Novorossiysk, but failed to clear the Soviets from the heights and coastal roads leaving it useless. Supplies would have to go to Rostov and down a few roads to supply the German advances.
Further to the east, the target was the Grozny oilfields. Grozny is the lowland, in front of the foothills of the Caucasus. Yet the German drive stalled, or was contained in September, in relatively flat territory, not the slopes (slightly) further south and west.
While it may be flat relative to the Greater Caucasus, the area north has few roads. After advancing for two weeks and 500 km, the heavily mechanized First Panzer Army was worn out. They had to be supplied with fuel, food, replacements, and spare parts over this increasingly long supply line stretching back to Rostov. German logistics were poor and not up to the task. The rapid advance and poor infrastructure made it increasingly difficult for the Luftwaffe to provide air support to First Panzer. The situation was made worse when some of their forces, including anti-aircraft, were diverted to Stalingrad leaving the Germans vulnerable to Soviet air strikes.
In 1941 major Soviet units allowed themselves to be encircled and destroyed. Now they were retreating in relatively good order continuously offering resistance. Every German advance took them further away from air support, lengthened their supply lines, wore out equipment, consumed valuable supplies, and cost valuable experienced troops. Eventually they were no longer strong enough to both advance and defend what they'd taken.