Though your root question doesn't fully have to do with ULX, as an admin mod, we do have many helpful features than can work with gamemodes.
First, your question.
Do some google research on the Source srcds command line (and, they can be included in Source autoexec.cfg too) options "sv_defaultgamemode" and "sv_gamemode"
These might be Gmod specific only, as the
very quick search I did seemed to only show Gmod.
Though, now thinking about it, I would hope other games use it such as Counter Strike Source, with all the different game modes CSS has available.
Now, on to ULX.
The simplest;
Our "ulx map <mapname> <gamemod>" command allows for changing of gamemodes when changing maps. Hardest part of that is knowing the exact gamemode name. (Some aren't so simple)
For instance, if I had a map called rm_track_and_field and gamemode that best worked with it called runningman, I could type
in console; "ulx map rm_track_and_field runningman"
Or chat; "!map rm_track_and_field runningman"
More advanced;
We provide map and gamemode specific setting capability in ULX.
Do this;
Go into your gmod/addons/ulx/cfg folder.
Copy the "maps" and "gamemodes" folders to your gmod/cfg folder
If you want (and haven't already), copy the server.ini from gmod/addons/ulx/cfg to gmod/cfg - then edit any settings in server.ini - they're well commented.
Then, read your gmod/cfg/maps/gm_construct.ini file.
and read the comment in gmod/cfg/gamemodes/sandbox.ini file.
Both of these allow you to have settings different for various gamemodes or maps.
Want <somemap> to ONLY play in <somegamemode> when it loads?
Using my example above, I'd create a file in gmod/cfg/maps called rm_track_and_field.ini
In that file, I'd add the line; "sv_gamemode" "runningman"
I might also turn off infinite aux power so no one could sprint all the time in that one map "sv_infinite_aux_power" "0"
(I don't remember if thats a valid Gmod command, but it works in HalfLife2)
"sv_gamemode" "runningman"
"sv_infinite_aux_power" "0"
Then, I might want to change the default running speed for my runningman gamemode
"sv_sprintspeed" "999999"
Hope this gives you the basic idea