Do I need to get a new AFSP approval if I change to a new part 61 instructor? I have an AFSP (TSA) approval to train with a part 61 provider. I want to transfer from them to an individual part 61 CFI. Do I need to go through the AFSP approval process again?
Yes, you will need to apply for AFSP approval for your new training provider. The Application Guide for the AFSP says in step 7:


  It is possible for a Candidate to have several active training requests at a given time. These requests may be for the same or different flight training providers. Each training request form will be processed separately; AFSP approval is valid only for the Provider listed in the application.


So if you needed AFSP approval for your original training provider, you will need a new approval to switch to a different provider.