Seven years ago, Dharamveer Solanki, a Hindu, left his home in Pakistan’s Hyderabad city, never to return. When his train crossed the border into India, Solanki said he felt happier than ever before.

“It felt as though I had been reborn,” he said, sitting inside a bustling refugee colony on the outskirts of New Delhi, where he and hundreds of other Hindus who fled Muslim-majority Pakistan have built a new home...