You will need the below documents:-
Transfer of Sponsorship Form duly filled and signed by yourself, your old sponsor and new sponsor
The signatures need to be verified at the Immigration
NOC letter from your Old sponsor
Police Clearance Certificate from your Embassy attested from MOFA
Old Company's ID Card copy, CR Copy & Municipality license copy
Educational Degrees attested from MOFA (optional)
NOC letter from the new company
New Company's ID Card copy, CR Copy & Municipality license copy

The documents will be submitted at Gate 4 at Gharafa Immigration. They will give you a receipt. You might have to wait for atleast 2 Weeks for approval. After approval, go to Gate 2 at Gharafa Immigration, pay the fee and collect your residency card under new sponsor. (this part is handled by new company's PRO)