You habe to check the website of the train compamy that run the train. Most international trains are cooperations and you habe to check for all operators.

The common rule is that small dogs in a box do not pay and big dogs habe to pay.

You need to check that on the website of the railway companies. The info should not be difficult to find. Just google dog and the name of the railway company. 

There are different rules for different countries and also different rules depending on the size of the dog.

The problem is I can find the fee for one-country routes but I can't find any rules for international, other than what Polish cashier told me about this particular train...

Travelling intermational between borders it should be 50% of the ticket price.

In most cases for INTern. it is simply the sum of what both countries on both sides ask for/charge. And yes-as above, it can indeed vary quite a lot between them. Also note that some countries also have extra rules about vaccinations or need to muzzle dogs-these are not set perse by the railways but by the local govmts.

 A simple google like:how to take dog on train in X / Y´ will mostly lead to answers. In some countries there are clubs of dog-owners who may also have much detailed info and tricks on how to lower cost.

Thank you everybody. I’m still not sure how it works.

Not really, because train I wrote about goes trough Poland, Czechia and Austria, and in none of this countries dog fee is ½ of normal human ticket, but in international train it is

We already sorted things out, unfortunately won’t take our dog children for the trip (they will however have all-inclusive stay with their human grandparents) and because of that have to shorten our travel but still - thank you for your genuine will to help and responses, it’s very nice community here <3