I have heard of this before. Mughals, persian word for Mongols, did have a bad reputation in their native Central Asia, because of the Mongol invasion and atrocities during the invasion. Although Chingisid descendants were the only one that were allowed to be Khan's, until very recently (Kumul Khan, 1930s and Khan of Bukhara 1920s), they themselves considered themselves assimilated Turks of Mongol ancestry. Timur was Gurkani (son-in-law), an Emir, but not a Khan. (These Khan's I think are different than most Khan's in South Asia)
I am not sure if it is Russian propaganda or not (Russians still hate Chingis Khan), but some Kazakh friends I met have told me about siege of Otrar, when we visited the ruins of that place:
THE HISTORY OF ANCIENT OTRAR
So Babur could be proud of Mongol ancestry at the same time not proud about the atrocities caused by Mongol invasion in that area, not sure if that makes sense.
I have interesting info about Persian influence on Mughal Empire, hope Shia's don't get mad with me. Both Babur and his son Humayun were helped by father and son Safavi, Shah Islmail and Tahmasp, both tried to convert them as a condition for their help and Safavi Iranians got access to Hindustan as a result. Babur and Humayun only pretended to convert, so that effort failed, but Shia became numerous in then Hindustan because of this history:
Chapter 4 – Humayun Restored
Facts about Humayun - DAWN.COM
And we have two history experts here in this forum, lets see if they can contribute:
@
Wholegrain @
ManUNITEDglory
@
Yzd Khalifa @
Arabian Legend @
al-Hasani the history in the links above may be interesting.