Pet Paperwork required once I am inside a Schengen Zone and going to another one directly?

I am living in US with my dog.

I will be travelling to Amsterdam first with my dog and will get all the paperwork from US and then get European pet passport once in Amsterdam.

My question is that if any pet paperwork is required when travelling to other countries within Schengen zone directly by train/car/ferry?

Edited to add more context.