Trains are considered to be very stable because they move over rails and because the radius of bends is really wide. The movement of a train, except in exceptional circumstances, is very smooth and maintains a constant cruising speed, they generally brake or accelerate in plenty of time to avoid sudden movements.
Which is why passengers rarely experience sudden jerks and movements while they are seated.
This is an important difference between them and road vehicles where there is a lot of uncertainty and unexpected maneuvers, or with planes that can experience sudden turbulence.
Trains do not have seatbelts or child restraining systems. Children are seated without any further requirements.
So, would it be sensible for trains be equipped with child restraining systems? The answer is a little complicated and is closely related to whether or not seatbelts should be installed. There is no agreement on this either technical or regulatory, there are people in favor and people against.
However, just like every other moving vehicle and if only to be prudent in anticipation of exceptional circumstances (emergency stop or in an accident), it wouldn’t be too much to ask that they were equipped with seatbelts or for children to use a child restraining system to avoid slipping down the seat or crashing into the furniture.
For children to use a restraining system on a train it wouldn’t even be necessary for the seats to have seatbelts, although if they did, it would be possible to use a car seat.
They could use the same child restraining system as they use on buses that have no seatbelts. These consist of a harness with straps that fix to the back of the seat. They can be easily dismantled and stored and take up very little space.
In this case, although not compulsory, better to be safe than sorry.