Promotion from ITA to Assistant Consultant

Hi , I completed 3 years in TCS and got 3 B bands in consecutive years. I have total 9 years of experience and still in C2 ITA.

I have been working in one BFSI project 3 years and recently switched to a Retail project and completing 1 month.

What is the process and criteria for the promotion to next designation and Band. Do I need to follow up with my current manager? As it is only 1 month in current project does it worth asking him now.

Could you please provide your thoughts and guidance on this please?