If you have a device driver that works on one Linux distro, will it also work on another one?
That depends on what kernel the other Distro is using.
Is the device driver in source code form, or compiled binary form?
If it's in source code, then it will work fine when compiled into the version of the kernel it's intended for. Compile it separately for each kernel configuration that you need it to work with.
If it's in compiled binary form, it is only going to work with whichever specific kernel it was compiled against. Even the very same version of the kernel, compiled with different configuration, will be unlikely to work with a binary module compiled for some other configuration.