To be extra clear I'm speaking about the period up to the Black Death. The Horde survived beyond that as an independent state - overlords of the oppressed natives. They launched further wars, but at that point it's much more a history of Rus as a European power, admittedly ruled by a foreign/Asian ruling class, but it's a different situation.

Let me give a longer reply that is also a broader re-consideration of the issue (having slept one night on it).

• I think one key factor for the Mongol withdrawal and more limited further action is imperial over-stretch. The Mongol armies tried to hold territory, and that territory got vastly bigger with each decade. Demanding tribute instead of direct occupation (as also in the attack against the Byzantine Empire) is another angle of that, from which follow wars against neighbours refusing to pay tribute. Note that in effect, the sack of Baghdad was a similar border war: the Abbasid Caliphate paid tribute earlier.

• Another issue is the separation of the Mongol Empire into smaller units, which started well before the end of the nominal joint empire. The Golden Horde started out as the region controlled by Batu and later Berke, who were in a minority faction. The area controlled by Hülegü (a member of the majority faction), Persia and later Mesopotamia, later became another independent state as the Ilkhanate. In other words, there wasn't a monolithic force seeking profit, but various factions all seeking their own profit with their own border wars, some bigger, some smaller. In fact, the faction that became the Golden Horde began to fight the one to become the Ilkhanate and ally with non-Mongol forces just two years after the sack of Baghdad.

• I submit the westward drive from the Ilkhanate is different from stuff like the destruction of Vladimir (but also the taking of the Song Empire) in the sending of auxiliary troops from the other sub-empires and the direct order from the Great Khan. However, the drive to Baghdad was apparently part of a broader westward drive, as Berke began another conquest of Europe (which ended when he fell out with Hülegü):

Golden Horde - Wikipedia, the free encyclopedia

In 1259 Berke launched savage attacks on Lithuania and Poland, and demanded the submission of Béla IV, the Hungarian monarch, and the French King Louis IX in 1259 and 1260.^[19] His assault on Prussia in 1259/60 inflicted heavy losses on the Teutonic Order.^[20]

_{*Lunatic*, n.
One whose delusions are out of fashion.}