I found this doc, which might be helpful for a user by user migration: https://docs.aws.amazon.com/cognito/latest/developerguide/user-pool-lambda-migrate-user.html You could run this and create users in the new IdP.
I found this: https://docs.aws.amazon.com/solutions/latest/cognito-user-profiles-export-reference-architecture/cognito-user-profiles-export-reference-architecture.pdf which is good for profile data, but explicitly excludes passwords. I suppose you could bulk migrate everyone using this approach and then require everyone to reset their password?
How do people migrate their user data off of AWS Cognito?
Full disclosure, I work for a Cognito competitor (link in my profile) and the doc I'm working on will be published on their site.
You said that you are working on Cognito competitor -- are you storing plaintext passwords in database? Is your service going to have user export containing passwords?
[0] https://docs.aws.amazon.com/cognito/latest/developerguide/am...
[1] https://en.wikipedia.org/wiki/Secure_Remote_Password_protoco...