The approach of different companies in this case is different.
In some cases, they agree to continue the employment as if there was no transfer or there is no break in employment. In that case you need not ask for a F&F as you will continue to get the benefit under Gratuity Act in the new company. For that, you must get the point clarified in writing from the management, specifically it should be specified in the new appointment letter you get from the company you move to.
If the company is not willing to give you continuity, then you should ask for full and final settlement before moving to the new company.