fpm is not a complete solution. It just creates a package from your files, however you need to build them in the environment of the distribution where it is supposed to work, with the same versions of dependencies. OBS is the best solution I know, but with it you need to write packaging scripts compatible with each distro you are targeting. It is quite time consuming and requires a good knowledge of native packaging tools.
You can also use any CI system that is able to execute builds in containers with your target distros. This requires a bit more scripting (just a bit), but modern CIs are easier to setup than OBS in case you need your own instance. This also allows you to use your favorite VCS and workflow you are comfortable with.