Transfer of shares to NRI by Indian Co

Yes ,You can issue shares subject to compliance of FEMA act,1999 read with FEMA(NDI) Regulations ,company has to comply with section 42/62 and also check that business + funding is within prescribed ceiling for auto approval or else you have to take approval from Centra Govt First (By applying proposal on DPIIT) portal ,reporting to RBI in FCGPR at Firms Portal with FIRC KYC through AD bank is Must ...also note at this juncture that ,shares alloted under non repatriable basis are considered domestic in vestment and FEMA doesn't apply .

you can issue shares by issuing shares certificate ,for demat in US ,it's happens through American Depository Receipt and for that listing in NASDAQ/NYSE required ,so demat will be local depository only ..

Disclaimer- not to be considered professional advice ,for educational purpose only

Thanks Chirag