I have a question I would like to ask that applies to mostly all Linux distributions (and maybe even BSD), but the answer will probably vary depending on what Linux distribution it is due to varying package managers, init systems, service managers, and so on. Is it in this case better to ask one question and expect multiple answers for different distributions, or ask multiple questions that ask the same thing but how to do it on a specific distribution?
This seems contradictory. Either the question "applies to mostly all Linux distributions (and maybe even BSD)" OR "[it will] vary depending on what Linux distribution it is due to varying package managers, init systems, service managers, and so on". It can't be both. I see three options:
The question's answer depends on the system used; phrase the question in terms of portability: I need to do foo, how can I ensure this will work on any *nix?
The question's answer depends on the system used; choose the system you are interested in: How can I do foo in BSD? And then ask multiple questions for different systems.
The question is actually applicable to multiple systems; just ask the question in its general form: How can I do foo in *nix systems.
I don't think it would be easy to ask a single question in this way, yet I'm also not convinced that only breaking it into multiple questions would fix the problem...
A question should be answerable in its entirety so deliberately asking a question that needs answering multiple times for different systems would (to my mind) be "too broad".
In particular you would be asking for answers derived from multiple separate skill sets so if you wanted answers for Linux-Debian, Linux-Redhat, OpenBSD, MacOS, ... then it's unlikely that any one individual would be able to answer for all systems.
The result of this is your question will most likely be closed as "too broad".
Too little effort
It's not strictly a rule but many users dislike seeing questions where the OP clearly didn't bother to research an answer before writing their question. Note the reference to "Search and Research" here: https://stackoverflow.com/help/how-to-ask
The major problem with questions similar to the one you describe in your question is that you've almost certainly put no effort in to research it for each system yourself. I say that because it would be unlikely to research each system and get stuck on exactly the same problem for each.
If you were to break your question into multiple (near identical) questions each about a different system, it would irritate other users and draw in the down votes.
What to do instead
What you should generally do in this situation is take some time to research an answer yourself first. If you get stuck on one then ask a question for that system. It may be that understanding from answers to that will help you with other systems.
It's fine if you get stuck on multiple systems, but try to space out your questions and avoid spamming near identical questions.