LEDs don't really have a forward voltage requirement. They have a forward current limit, and that in turn affects the voltage. Current limiting is most commonly done with a resistor, the value is chosen with simple math using the forward voltage drop and desired/maximum current. You can use almost any voltage you want until you reach the sparky levels.
However, the above only applies to raw LEDs. if you are using a manufactured product you must use whatever power the product calls for. If it wants 5V, you buy a 5V supply.
5V is standard USB bus power, so if the product is intended for use anywhere near a computer they will probably build it for 5. 12V is standard in vehicles. Both voltages are popular away from those markets, with the choice often as boring as it matches other things that company makes.
These are constant current LED luminaires which must be matched to a specific driver circuit, which is the thing you call a "transformer" which was bundled with them. It has the capacity to drive one LED luminaire. That gives you no choice in the matter. Each luminaire must be used with a matching driver module.
It's as simple as that.
The good news is these luminaires appear to be "dumb" with no electronic components onboard, that's why they need such a particular driver. Electronics tend to fail not LEDs proper, so you won't have to replace the luminaire, only the driver.
18 AWG intercom/thermostat wire should suffice if you need a longer run between luminaire and driver. Thinner wire isn't really any cheaper, as this size benefits from economies of scale.
Now, what about using a different LED driver so you don't have 10 driver modules?
Being dumb LED emitters without onboard electronics, their spec is very clear that they must be driven constant-current. The only sane way to drive multiple loads constant-current is with series wiring.
In series wiring, current is exactly the same through each device, but voltages add. Current will be 320ma, or 0-320ma if you want to dim them without PWM. Voltage will be 20-30V x 10, or 200-300V.
Here's the problem. As a practical thing, that's a really goofy spec for a driver. I don't think you'll find one on the market. I could be wrong.
Also you're dealing with rather high voltages. THWN or NM cable will suffice, but they no longer qualify as "low voltage" under NEC rules. You'll need to follow all the formalities of proper Code wiring, and may run afoul of rules pertaining to voltages over 250V.
Also, DC voltage that high is one nasty piece of work. It is as aggressive as AC power 10 times its voltage, indeed, many switch and relay makers derate 80-90% for DC. You will not find a switch which will interrupt it without arcing, short of the spring-action "snap switches" used in very old homes (being vestiges of the DC age). Once it starts to arc, it will not know pity or remorse or fear and will not stop unless killed. The driver would contain it to 320ma, but you'd want arc fault detection in the driver. I would at least want to run that in steel conduit.
Best Answer
As @digitalninja mentions, this is a constant current led driver. Unlike a variable current/constant voltage power supply, that attempts to keep the same voltage output regardless of the current pulled, a constant current supply will attempt to keep the same current, by varying the voltage.
It will, given a load like an led bulb, change the voltage until the current becomes 350mA. That could be a 24V 350mA led bulb/strip, or a 12V 350mA led bulb/strip, or whatever.
As such, you are looking for bulb that states it is that current draw within that voltage range.