Train tckets: stopping along the way

Yes, you can do it if it's not ticket for specific train. In some countries if the ticket is for a longer distance is usually valid for 3 days but you must start journey on the date printed on the ticket. I am not sure if this is the case between France and Switzerland but any railroad info-center in France will be able to answer your question.

I believe that you can do this in Switzerland as long as you are doing this on the same day and you are on a regional train or a train that does not require seat reservations. Best to buy the ticket in Chamonix and not online. Just be sure that you are on a train out of Chamonix-Mt. Blanc that is going to Geneva via Martigny and Lausanne. At least half the trains from Chamonix to Geneva go through France and thus will not stop at Montreux or Lausanne.

If your ticket has a seat reservation or a specific train listed, then you have to take that specific train. Other tickets, however, are valid for that route for the whole day or for a certain period of time (6 hours on the Italian regional trains, for example - once you validate it your time starts). The tickets I've used inside Switzerland have mostly been just for that route so in that case I believe you could get off and get back on later. However, the exception has been when I am taking a train from a different country (Italy in this case) into Switzerland - my ticket was for that specific train and I would not have been able to get off.

Italian regional train rules changed a while ago. The validity of regional tickets from the time stamp on the validator is conditioned to distance.

In Germany, unless indicted otherwise, a standard (full) fare ticket on a from point A to point B is valid for travel on the route shown on any day during the period of validity (30 days?), until the end of that day for trips of less than 100 km, until the end of the next day for trips over 100 km. But if your ticket is from Frankfurt to Munich and shows that it goes through, say, Stuttgart, you cannot take a route through Nürnberg instead.

For instance, last September I had a ticket from Frankfurt airport Fernbahnhof by way of Karlsruhe to a small town in the Black Forest. The ticket was for about 2 PM. Due to the arrival time of my flight, I missed the first train. The next connection, 30 min later and more expensive, went through a different station than Karlsruhe. The only way I could take that train was to exchange the ticket and pay the difference. However, there was another connection through Karlsruhe an hour later. I took that train and the original ticket was accepted by the conductors.

However, you can extend the time duration by using the stopover option on the Bahn query page to build in extended stopovers, even overnight, when booking. I've seen connections, with stopovers that, as booked, finish later than the end of the second day.

In the case of advanced purchase, discounted Saving Fares, however, the ticket is specific to the train (date and number) shown on the ticket. If you miss a specified train, the ticket is void, unless you missed the connection due to a fault (e.g., late arrival of a train) of the Bahn. In that case the Bahn will allow you to take the next available train, even a higher class (for instance, an ICE instead of an IC). Best have the ticket office verify the missed connection and specify the next train to take.

Reservations are independent of the ticket. I had a reservation for the first train to Karlsruhe. When I missed that train the reservation became void but the ticket was still valid. I could have purchased a reservation for the next train before it left FRA.

As for Chamonix to Geneve, at least make sure the train you ticket from Martigny is one that stops in Montreux and Lasanne.

But why worry about it now? Individual tickets from Martigny to Montreux, Montreux to Lusannne, and Lusanne to Geneve cost only about $6 more than Martigny to Geneve, so be prepared for that much and ask at the ticket counter if you can make extended stops on one ticket for the entire route.

Looked up the Trenitalia explanation on validity for regional trains - for trips under 200km, the validity is 6 hours. For journeys longer than 200km, the validity is 24 hours. The exception is in Lombardy, if the trip is under 50km it is valid for 2 hours but that only appears to apply to that region. It also says if you are on the train when your validity expires you can continue to the destination station but may not take any stops in between.