The Mughal empire did NOT have this piece of land in the past and modern India doesn't even have the full stretch (Kashmir, Pakistan) under independent "India". History lessons are sorely in need for Indian leaders here. This land is NOT your's and in fact it has never been at any time during history.
So before the Mughals, there were periods of competition over Ladakh and Xinjiang between all surrounding power centers. Remember, the Mughals were just the last in a long line of Muslim dynasties that ruled India. Even the Ummayads had a presence here, and there was some contention over Ladakh between 3 different power centers:
"Fisher, Rose & Huttenback, Himalayan Battleground (1963), pp. 12–15: "Ladakh's geographical position leaves no room for doubt that its ancient caravan routes must have often served as a path first for conquest and then for retreat of the opposing armies as they alternated between victory and defeat."
There were periods of competition, but mostly trade/cooperation prevailed. Most of the history of this rocky region is actually peripheral to the general relations between the power centers around that area. This is why historically, this region has been mostly peaceful, because relations between China and the Muslim Dynasties were mostly peaceful, focusing on material and scientific trade. This is why one of the first things that Pakistan did upon independence, was give China back its claimed land in Kashmir. Because there was never any 'crusade' between our two peoples. Our actual nemesis has always been European.
In any case, the point is this: Peace in this region is a function of competition between the power centers around this region. Just because in history this area was mostly peaceful, that doesn't mean peace is the natural state of this region. It was only peaceful because the relations between those power centers were peaceful. Not because the area itself has no value or has been unclaimed. Therefore, whenever competition returns between the centers of power, so will conflict return in this region.
Granted, "India" had nothing to do with this back then. It has no actual "historical claims" because it has no history. But that is besides the point. Competition in this area has returned, because the players around the area are competing again. As for the question of "why" competition returned as soon as "India" came into being, well that is a whole other story. Maybe it's because they were ruled for so long that as soon as they got any power they went aggressive... Or maybe they were just always like this. What we do know, is that the historical records of the dynasties who have ruled them, are not very flattering towards "Indian" tribes in general. Neither the British, nor the Muslim dynasties held most of their local tribes in high regard, especially the Northern ones (the ones who think they are the "Aryans" lolz.)