How much fusion do you allow after which some people stop being considered 'heirs' to a particular empire? 100 years? 300 years?
Firstly no Hindu claims to be heir to the Mughal empire. We claim the heritage to the extent that it is a shared heritage (the influences in architecture, food, music etc). Of course, we cannot also close our eyes and minds to the numerous monuments, which were built with the sweat, blood and tax money of our ancestors. And not just my ancestors, but also the ancestor of every Pakistani on PDF.
I hope you know that the Mughal empire stretched till the Bay of Bengal through a combination of administrative machineries. There are various ethnic groups that inhabit these lands - Hindu jaats, Rajputs, Biharis, Bengalis and Assamese. That is why I asked first - how much fusion do you allow? The Mughals were numerically a small group when they arrived in India. Pakistanis would conveniently restrict it to a level where their fair skinned brethren can be passed off as a Mughal, despite the fact that early Mughals had distinct Mongoloid appearance.
What is referred to as Persianized Mughal court culture existed in large parts in an area that was known as Central Provinces and United Provinces during British rule. Some Muslims from this region (as well as others) migrated to Pakistan after 1947, but a large majority stayed back. Do these Indian muslims have more right to claim the Mughal empire?
Lahore was not the only seat of the Mughal empire. Delhi and Agra were also Mughal capitals at different points. For conquerors, a battle for India was the battle for Delhi, not Lahore. This is not to deny the importance and cultural majesty of Lahore, but one needs to call out revisionist tendencies, whether they are of the Pakistani variety or Indian variety. There is a lot of attempt in India these days to resurrect Hindu figures and events from history (and I have absolutely no problem with that), but in the process we are giving Pakistan an opportunity to act as the sole custodian of our Mughal/Islamic history. Even if BJP is trying to shred those links, the claims of heritage cannot automatically fall on the lap of Pakistanis exclusively.
Frankly all claims to glorious empires are hollow in today's times. The countries that have done well are those that have chosen to focus on the present and future.