Is there a check in and passport control procedure for international trains in mainland europe?
I’m going to be travelling Amsterdam to Hamburg, Hamburg to Copenhagen, Prague to Bratislava, and Budapest to Split by train.
All of these involve a border crossing. Is there a ‘check in’ process?
Is passport control done at the departure station like with eurostar or at the border crossing?

I have traveled back and forth from Amsterdam to Germany many times (with German ICs or ICEs) and there has never been a check-in-procedure at departure. The train departs as any other train does too, so no need to be there early! However, there were passport and sometimes also customs checks (especially regarding drugs on the train from The Netherlands into Germany) at the border (the first stop after the border), but not always. The procedure takes place as the train rolls along, so there are no delays involved. I have also been on the train from Hamburg to Copenhagen, and again there was a passport control at the border. As for your other routes, I don’t know about Eastern Europe, but I would assume it’s the same. Have a great trip!