How did Brahe get to Prague?

The river Elbe featured in Tycho's travel plans, but not as a means to travel on it, but mainly along it. It was with lots of carriages for all his entourage, family and instruments, in various configurations, and in multiple stages — back and forth — between Wandsbek, Hamburg, Magdeburg, Rostock, Wittenberg, Dresden, Wittenberg and finally Prague…

The episodes from Dresden and Wittenberg were not really deliberate on Brahe's part, as the emperor isolated himself in paranoia fearing epidemic contamination, thus slowing down communications at first then sending Brahe away for a time.

Initially he wanted a slightly more direct journey,

Tycho's decision to move in two stages may also have been influenced by the logistics of moving. This seems a bit unlikely, because both Hamburg and Prague were on rivers, and water transport was the most efficient way of getting sizable loads from one place to another. But the possibility of boat or barge travel seems never to have figured in Tycho's plans. From as early as January, when he made arrangements through Rosenkrantz to buy a used travel coach in Denmark, Tycho had obviously decided to move his entourage not on the river itself but, rather, on the roads alongside the river. […]

In letters written on 5, 9, and 14 September, Tycho inquired urgently after the coach, stressing that the need for it was all that stood between him and departure for Prague. On the twenty-fifth, finally, word that it had arrived at Lübeck. Four days later, on Michaelmas, Tycho left the last quarters in which he would ever be settled for even as long as six months.

In what might have been an impromptu change of plans, the whole Brahe entourage started the journey. It must have been a formidable enterprise. In his impatience over the delay of his coach from Aarhus, Tycho had finally purchased another one in Hamburg. Including the one he had brought with him from Hven, as an indispensable badge of rank, there were three coaches in the train that must have included several baggage wagons and at least a few people on foot and horseback. For the first leg of the journey they traveled some two hundred kilometers to Magdeburg.

[…]

The Brahe men, accompanied at least by Tengnagel and a retainer by the name of Andreas, continued south along the Elbe until they reached Dresden. At that point Tycho sent a messenger to Prague to inquire about the protocol for approaching the emperor. Whether this was a standard procedure or a special precaution, it was well advised; for the dirty crowded capital of the Holy Roman Empire turned out to be in the grip of some kind of epidemic, and Rudolph had scurried to the countryside to seek safety in isolation. So effectively had he sealed himself off from possible contamination that it took a month even to get Tycho's communication to him an answer back out.

Given the extremes to which Rudolph had gone to protect himself, his response was a foregone conclusion: Tycho should stay a safe distance away and wait for the emperor to call him when the plague abated. With a few words therefore it was ordained that Tycho would spend the winter in neither of the places in which he could reasonably expect to find comfortable lodgings for the entire family: Wandesbeck was too far away from the emperor, and Prague was too close.

— Victor E. Thoren: "The Lord Of Uraniborg. A Biography of Tycho Brahe", Cambridge University Press: Cambridge, New York, 1990. (p381–446, quoted obviously in short form here, but really a hilarious read… One wonders when he ever had the time to unpack and gaze up…)

Small clarification on place names and terminology: Brahe left 'his' island Hven for Wandesbeck in October 1597 after King Christian IV defunded him. When Brahe started this epic journey from Wandesburg in Wandesbeck/Wandsbeck/Wandsbek near Hamburg, this was still 'in Denmark'/Danish at the time.