India didn't
inherit Kashmir from the Dogra King. Pakistan and India was carved by British based on Radcliffe line.
It was the Sikhs that for the first time in perhaps millenia rid North West India of Foreign/Islamic Rule. Sikhs not only conquered Afghanistan's summer capital Peshwar, but fought and defeated Kishtwaris, Ladakhis, Tibetans/Chinese, Dogras, Gorkhas, Multanis etc.
Ladakh was won in 1834. Maharaja Ranjit Singh had then stopped Gen Zorawar Singh from pressing on on that front as the Chinese emperor was much stronger. This meeting happened in in 1836. The flag that Zorawar Singh fought under was the Sikh flag btw.
However, after Maharaja Ranjit Singh's demise and after Gen Zorwar Singh had conquered most of regions that now constitute J&K, he decided to again try and expand the empire to Western Tibet. As mentioned in
@Ray sir's post.
British had then sent Cunningham because they were afraid of the blowback to recall Zorawar Singh from his Tibetan campaign. Maharaja Sher Singh (of Lahore Durbar) acquiesced after being evasive for sometime. But before the message reached General Zorawar Singh to pull back he was killed in battle.