When I take a look at the firmware github repo, I can download kernel images. But to what distro do they belong to or are compatible with?
Firmware image contains couple of components. There are couple files that creates bootloader (start.elf, bootcode.bin, loader.bin) and couple of kernel images (normal, emergency and cutdown). There is no source available to bootloader files so every Linux distribution is using the same files. They may differ in which versions are they using.
Kernel, on the other hand, is distributed both in binary form (compiled) and source form (you can compile and modify it yourself). Binary kernel should be compatible with all distributions but some features that particular distribution is using may not be enabled in it. This is why some distributions build their own version of kernel (with modified config, some patches added etc), based on the official source files and use it instead. Some of them may be even using some other kernel version that has enabled support for RaspberryPi basing on this official sources. So it's up to distribution maintainers to decide if this is needed or not. For example raspbian distribution (and most of the others) is using official kernel without any modifications.
You can download the image from github and boot from it (just remember to download modules too). And if you are using rpi-update tool, you are using this kernel/bootloader files too.