GR has NO trains over the border at all.

For overtheborder direct trains (cost also an awful lot extra) best to do early-these very often give probs.

The rest; depends-some routes are very populair, other much less so (f.e. in spring this yr I found whilst in FR several such less-wanted TGV routes a day ahead still with the small 10€ quota open). On lines with trains ev hour or even more-with a little flexibilty-avoid peaks-its often possible even on the day itself.

It's best to check timetables in advance so you can make a good travel plan. Trenitalia has not yet released their timetables for most of September, so you'll have to wait a bit. Although you can check timetables on an earlier day (on the same day of the week) for which timetables have already been released to get an idea of what's possible.

September is less busy than summer, so booking a long time in advance should often not be necessary, except for the international TGVs.