The phrase in announcements for these trains will be e.g. "This train divides at Southampton. Passengers for stations to Poole should be in the front 5 cars of the train".
Because (as described below) these trains are actually constructed from Multiple Units, the train can only be sub-divided if it in fact consists of two or more smaller trains, for example 450s come in multiples of 4 so a 12 car train could divide into three. Some designs offset the driver's cab so that a gangway for passengers fits through the "nose" of the train when it's connected. In other designs you must leave the train and walk along a platform to pass the point where the trains are joined.
On one UK journey from Portsmouth to Preston (Virgin Trains on Sundays), the train would divide en route and the rear half continue to Preston and the front half would go to Manchester.
One time, the conductor announced - several times during the first part of the journey - that due to an electrical fault in the rear half of the train (no lights), that part would go to the depot in Manchester, and the front would go to Preston, so when the train stopped for splitting, please would everybody swap carriages and make sure they were in the right half.
Needless to say, a few minutes after our section was on its way to Preston, a few fretful discussions that included the phrase "...didn't you hear the announcements?..." could be heard, followed by the odd expletive.
On another occasion, my train left London Victoria for (I think) Brighton, Bognor Regis and Southampton Central (a double split to make 3 x 4 coach trains), and the announcement went something like...
"This is the xxx train to Brighton, Bognor Regis and Southampton Central. Due to a fault, this train is composed of 10 coaches instead of 12. This train divides on route so please make sure you are in the correct part of the train; the front 4 coaches will call at....the rear 4 coaches will call at....and the middle 4...oh we've only got 10..oh, how are we going to do this...erm...hang on I'll have to have a think about this...."
Southern has a bunch of different variants of class 377 in different lengths, so presumably your 12 cars would have been 3 x 4 car length 377s.
Depending on what they had spare ("due to a fault") the 10 cars might have consisted on 2 x 3 car plus 1 x 4 car, or of 2 x 5 car configurations.
In the latter case obviously the train can't split three ways, that's impossible. I would expect that for your service they would choose to run the train to Brighton, then split it and run half to Southampton Central, half to Bognor.
To be fair this route actually could be served (albeit with a little delay) by a single train, there's no huge divergence along the route to serve all of them, it's mostly that it'd be a pain to get back out of Bognor heading for Southampton, the signaller may not want to allow the relevant movement (crossing from the up to the down side, or reversing on the down side) and the driver may not be trained to do it even if a signaller will signal it.
In the Netherlands we have trains with the cab above the passenger compartiment[1][2] but the national rail service has recently revised these and welded shut the passenger walk-through coupling due to maintenance costs and frequent breakdowns of the many moving parts.
Because (as described below) these trains are actually constructed from Multiple Units, the train can only be sub-divided if it in fact consists of two or more smaller trains, for example 450s come in multiples of 4 so a 12 car train could divide into three. Some designs offset the driver's cab so that a gangway for passengers fits through the "nose" of the train when it's connected. In other designs you must leave the train and walk along a platform to pass the point where the trains are joined.