Going from one country to another is normally one train company, from the origin country, and then moving on from that city - in another country - means a different company, from that second country, and so on.
The situation in eastern Europe is more variable, but the reservations aren't generally as prohibitive as in France.
The neverending "refugee crisis" (the official reason several countries have introduced checks within the Schengen area) has allowed more frequent checks, but it seems fairly random whether or not they happen. Out of perhaps 10 overnight journeys, I've not been checked going south or east, but around half the time I've been checked going into Denmark from Germany.
It's so disruptive that I no longer take the overnight train into Denmark. You should prepare for it — put the passport in the mesh holder so you can just hand it over — but I find it difficult enough to sleep on a moving train anyway.
Having recently visited Germany via plane, they have certainly gotten stricter/performative. Leaving Munich for London and despite already have presented passports to immigration, they had police checking passports on the gangway to the aeroplane…
Likewise a delay from Berlin to Geneva (only two countries, but a long trip) and again no problem with my re-route. Ended up 30 minutes late