Need guidance to transfer stocks from US based Broker to India based Broker

Dear Grapeviners, I need your guidance. I was working with a company listed on NYSE and I had enrolled in their ESPP program. The stocks allocation and management was being done by the company sponsored US based brokerage firm. Since I am not the company's employee anymore, the company has provided a temporary extension to the sponsorship as per their policies which is going to end soon. I want to transfer these shares to an India based brokerage firm. Hence I am looking for advice and suggestions if anyone has done something similar in the past; especially in regards to things to take care of before, during or after the transfer. I am also looking for recommendations for brokerage firms which are reliable and provide good customer support. Many thanks in advance for your support. P.S. If possible please don't suggest new age brokers like Groww or Zerodha as their customer support is not up to the mark from what I have heard.