DKMS modules for multiple kernels

I have a Ubuntu Studio 16.04 installation with both generic and lowlatency keels installed that I can choose when I boot (I generally use generic but I may use lowlatency in case I want to do something with Jack, for example). I've noticed that DKMS modules (in my case, the driver for my Nvidia GPU and the VirtualBox module) only gets built for one of the keels - I think the one I'm ruing. Similarly, update-initrd is only run for one of the keels - I think the latest it can find in /boot in alphabetical order. So, for example, a few days ago I had to manually run update-initrd for the generic keel after updating the GPU driver to be able to have graphical output; meanwhile, the lowlatency keel boots fine, but it is not loading the Nvidia driver (it is using i915) because it has not been build for that keel.

So the question(s) would be:

• Is there any way to rebuild a module for a particular keel? Even if it is by hand, I mean just a sequence of commands I can run to do it. I already leaed how to use update-initrd to avoid the problem with initrd.img, but not how to recompile the modules.
• Isn't there any configuration I can tweak to enforce this to happen automatically? Like, some configurable script or logic that decides which keel(s) are considered for these tasks.