From what I know, Mughals did adopt local customs, or at least the ones which were worth adopting. They weren't going to adopt widow burning obviously, but not everything will have been bad, I am sure they learned from the locals same as the locals will have learned from them.
The famous Widow burning jab. It was never wide spread and limited to sporadic cases unless they were attacked by Foreigners and forced into slavery.
Anyways, Mughals spoke Farsi, wore Persian attire and ate Turkish/Arabic cuisine for the most part. Their Indian cooks did fusion of Indian and Turkish/Persian cuisine in their kitchens. They married either Rajput women or imported their wives from Persia and Arabia. They lived in their plush palaces never stepping out of it.
Mughals remained foreigners who were occupying the land by force. No wonder the locals were always plotting against them and even sided with British to dethrone the Mongolic Mughals.