Their Hungarian episode might be an interesting addition to the Eastern European history of the Teutonic Order.
Before they moved to the Baltics the order briefly settled in Transylvania to fight and convert the steppe nomads, mostly the Cumans.
Hermann von Salza was the Grand Master then, whose family was vassal of the Thuringian Landgraves. The Arpads had good connections to Thuringia and king Andrew II who led a crusade to the Holy Land beleived its a good idea to settle a military order to the sparsely populated borderlands of Hungary.
In 1211 the knights arrived. They were donated privileges like exemption from taxes, monopoly of mining and trade and Hungarian church hierarchy. They had an important role of organizing German immigration to Transylvania. By 1222 the had five strong stone-castle (like Marienburg, Rosenau etc.). They successfully repelled Cuman incursion and conquered their lands.
In 1224 the Grand Master offered direct overlordship of the Order's lands to the pope. Honorius III. accepted it and with this act the "sovereignty" of the Hungarian kings practically ended. Naturally Andrew II couldn't tolerate this. With a quick campaign he captured the castles of the Teutons and ordered the knights to leave Hungary. He stripped them of their lands. The Teutonic Order had to find a new home and they found it in the Baltics.
They had an other even shorter episode in Hungary. In 1429 king Sigismund (also king of Germany) offered them the banate of Severin to protect the borderland from the Ottomans. Their leader Nikolaus von Redwitz was appointed as ban. But three years later they suffered such a serious defeat that they had to leave the kingdom.