Indian assembly yes.
Some parts yes right now the Wings are planed to move there.
As is of course the Wings of F16 were already made in Isreal and South Korea.
Tata has to certify that they can produce the Wings to spec. That's about 2 years from now.
Right now though if you look at a parts manufacture list for F16 Block 70/72 It's an atlas.
Production also depends on the contract and willingness to source.
Bahrain will be the first customer from the Greenville plant. Likely followed by Slovakia.
Long term yes Lockmart wants to grant Indian production which might mean South Carolina as a second line for F35 or evetually F-X/F/A-XX perhaps KC-Z or the eventual C17/C130J replacements.
but that's likely to be around the turn of the 2030s.