What was the fighting style and equipment of 13th century Eastern European Baltic and Slavic cavalry?

The Livonian Order or Frates Militiae Christi was founded by Albert von Buxhoevden in 1202. The stronghold was around the area that today is Latvia and Estonia and have permission by Pope Innocentus III to defend the values of the church both earthly and divine.

They have some similarities with the Knights Templar and the Teutonic Order. The later absorbed after some years the Livonan Order into its organisation as an autonomous branch of the Teutonic Order.

Overview

This means we are thinking about the Baltic tribes—modern Latvians, Lithuanians, and Prussians—as well as the Slavic peoples—making up Kievan Rus, Novgorod, and other principalities.

While the Mongols, on whom one of the other answers focusses, were indeed cavalry based, I think it is slightly misleading to include them as a culture of Baltic & Slavic Eastern Europe though without a doubt their advances had very great effects on military development in Russia.

Also, both of the presently-existing other answers mention the Sword Brethern / German Order. This is a good inclusion (though not native to the area, the Germans made the Baltics their homeland), but it the primary method of Crusader warfare in the Baltic was holding and fortifying outposts. If questions about the German knights are outstanding, these should probably be treated in a different question.

Latvians

The Latvians seem to have kept to their 'traditional' style of war though one thing I don't see mentioned below, but think I remember from a different source is that the horses were quite different to the Western European warhorses and smaller though sturdier, which probably helped them over long and difficult journeys.

The most distinctive characteristics of Baltic crusading warfare reflected the harsh climate, difficult terrain and the unorthodox tactics used by their pagan foes. ... Summer operations tended to be on a larger scale, often employing seasonal Crusaders and relying on river or naval transport. Small-scale Crusader raiding was undertaken in winter by resident troops using frozen rivers and marshes as highways. Such winter operations also led to a heavy loss of horses.

Indigenous Baltic military skills concerned rapid raiding by mounted infantry and the laying of ambushes. Surviving descriptions of such raids make the local warriors sound remarkably like the woodland Indians of North America. The cavalry elite of Estonians and Balts "rode in the ancient fashion", according to their Crusader foes, not using the tall wood-framed war saddles of western knights. Their battle tactics had much in common with those of the Mongols, except that Baltic hors*m*n used javelins rather than bows. They would charge their enemy, hurl a javelin, then retreat, repeating this until all javelins were spent.
_{—Nicolle, 'Lake Peipus 1242'}

Lithuanians

I've got a description of the 13th century Lithuanian fighting style in this answer here which should be read alongside the above quotation.

Russians

Militarily more important ... were Turco-Mongol warriors of steppe origin. Their influence on Russian armies was enormous. Since it proved impossible to make Russians into effective, or at least numerous, horse-archers, Russian rulers constantly recruited steppe peoples for this purpose and as herdsmen, to raise the horses needed by the druzhinas. The flow of such specialists into Russian territory was helped by a Turco-Mongol tradition whereby the military elites of steppe tribes often migrated west or north if defeated by newcomers from the east, while the bulk of their tribe was absorbed by the conquerors. Various waves of such defeated elites formed the famous Chernye Klobuki or 'Black Caps', who played a major military role in the late 12th and early 13th centuries. Most were found in southern Russia, but their fate following the first Mongol invasion is unclear. ...

Ancient infantry traditions survived, but Russian cavalry had also developed in the 10th century in response to the threat posed by steppe civilisations to the south. This included a large number of light cavalry, including some horse-archers drawn either from the steppe nomads themselves or from steppe influenced borderland Russians. During the 12th and 13th centuries a disciplined cavalry elite was modelled on that of the Byzantine army, armed with close-quarter weapons. Even so cavalry as a whole was more typical of southern and perhaps central Russian states than of northern Novgorod.

Armies of the states of Kievan Russia were generally in three parts: the princely druzhina; urban militias that had largely replaced the old Russian tribal levies; and assorted non-Slav auxiliary or allied contingents. The druzhina itself evolved as a result of wars with the steppe nomads. Bound by oaths of loyalty to its knez, the druzhina was highly mobile, well trained and well equipped. Two levels of druzhina developed in the 11th century: a senior druzhina comparable to the 'household' of a western European prince, and a junior druzhina consisting of armed retainers. _{—Nicolle, 'Lake Peipus 1242'}

These Mongol-inspired horse archers are also interpreted as having dealt the decisive blow at Lake Peipus.

The Rus mounted warriors, including the influence of the Chernye Klobuki, are described as primarily steppe-originating though with some European influence (both from Constantinople and the Baltic trade routes):

Russian noble warrior: Relations between the Chernye Klobuki and Rus' ruling classes were close. Mutual influence between their military equipment is seen in the short mail hauberk and new form of cuirass worn by this Russian boyar. His helmet, weaponry and horse-harness are, however, more European. ...

Senior member of Chernye Klobuki: The main feature which distinguishes this 'Black Hood' leader from the boyar is his archery equipment and lighter armour. He uses a whip rather than spurs, rides with shorter stirrups, and wears a helmet of Asiatic form.
_{—Nicolle, 'Armies of Medieval Russia, 750–1250'}

Meanwhile, in focussing on a slightly later period, these changes are brought out which overall indicate an inclusion of Mongol traditions, but also the incorporation of more European armour into Rus armies:

Western Russian cavalryman, fully armoured: This horseman's arms and armour illustrate the mixture of military influences seen in western Russia during this period. The helmet is a type also seen as far away as the Byzantine Empire and the Balkans, while his 'grooved' or 'keeled' shield is of the so-called 'small Lithuanian pavise' type. He is armed with a spear and javelins rather than spear and bow, indicating that Lithuanian light cavalry influence was stronger than that of the otherwise dominant Mongol-Tatars. The sword was probably imported from central Europe.

South-Eastern Russian cavalryman: This fully armoured horse-archer, from that part of Russia most exposed to Turco-Mongol military influence from the steppes, has the abundant military equipment long associated with the military elites of these regions. He does not, however, wear lamellar or any form of armour other than a simple short-sleeved mail hauberk. His archery equipment and curved sabre are similar to those seen across south-eastern Europe, much of the Middle East and as far away as Central Asia. A small hardened leather wrist­ protecting bracer was often worn on the lower left arm.
_{–Shpakovsky & Nicolle, 'Medieval Russian Armies 1250–1500'}