First thing to know about this "swipe between full-screen apps" is that in terms of functionality, this is exactly the same as changing the
Space (A little more on misson control and spaces here). Fullscreen space is just a type of space that contains just one one window/application.
Some mouse software may have a built in "action" that might be called "Move left a space" or "Move right a space", but that is not guaranteed. However, pretty much any mouse software can set all or some of the buttons to trigger shortcuts and luckily there are native shortcuts in Mac to go to change spaces:
- You can move left one space with the shortcut Ctrl + Arrow Left.
- You can move right one space with the shortcut Ctrl + Arrow Right.
At the bottom of this answer I wrote where these can be changed or enabled if necessary. You can also go directly to a specific space: You can see "Switch to Desktop..." in the screenshot at the bottom.
How you can trigger these shortcuts (or actions) will depend on the mouse and its software. Many mice, especially gaming mice, have a Mac software even if they are not advertised as Mac mice. So if the mouse software can set shortcuts, then you're all set. There's nothing I can say about how that works exactly without knowing which software your mouse would use, but typically those are fairly simple: Pick a button and remap it to a specific shortcut that changes the space.
If for some weird reason your gaming mouse doesn't come with a software capable of adding a shortcut to a key or it's terrible, there are still many 3rd party applications that can do it.
Here's a few 3rd party apps that you could try:
It's not necessary, but I'm pretty sure both of these have their own built in "Move right and left a space" actions you can remap to buttons.
Additionally there is the "Mission control" shortcut, which is by default in the F3 key in your keyboard. When you click it, you can pick a specific space from the top of the screen using your mouse. If that works in games, which I think it might, if you have multiple spaces, it might be the best way to do it.
You can change the native shortcuts in:
System Preferences →
Shortcuts (top) →
Mission Control (left panel) →
Move left a space / Move Right a space (right panel)