Just wondering, if I had a question that wasn't getting answers on mature sites such as Stack Overflow, Server Fault or Super User, could it get migrated to a beta site?
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Migration paths won't be defined until sites are out of public beta.
In the meantime I recommend deleting your question and re-asking it on the public beta, if you feel strongly it's a better fit there.
There are actually three issues in your question.
Let's start with the premise that migration is a way to address not getting any answer on the original site (regardless of beta status). By Stack Exchange policy, only questions that are off-topic on the original site may be migrated. The rule exists for a good reason. Many questions fall within the scope of more than one site, and allowing migration based on the sentiment that a question might do better elsewhere could lead to questions being moved around on a whim. Also, if questions are routinely migrated from low-traffic sites to high-traffic sites, the low-traffic site would never get a chance to mature.
Next, think about why the question hasn't received an answer. It might be that it simply slipped through the cracks. It could also be that it has quality issues. It's also possible that it's a difficult question that isn't easily answered. Judging the situation based on the votes, comments, and view count, you may need to edit the question to clarify it, add detail, or limit its scope. A bounty might be appropriate. Note that reposting or moving a problematic question just to draw attention would not be appreciated by others.
As for the question of whether a migration is possible: yes it is. Moderators can migrate questions to any site, even if it is not one of the standard migration paths. If a question needs to be migrated, just flag the post for moderator attention, and write your request in the box. You should explain why it's off-topic on the original site, if it isn't obvious.
That said, if you want to ask your question elsewhere, you can just do it yourself. You can also optionally delete your original question if you haven't received any answer on it. If you decide to leave your question on both sites, please declare your cross-post with mutual links as a courtesy to other users.
This question was asked in 2010 when there were many fewer sites and the lines between them were clearer. Things are different in 2016.
I agree with most of this answer -- moderators can migrate to anywhere but are unlikely to do so if the question is on-topic where it is. You are, therefore, better off re-asking your question on the other site.
However, I don't think you need to delete the original one. Cross-link the questions to be sure, but your questions should be different, each one tuned to its site. Sites have overlapping scope, so a question can fit more than one site, but chances are very good that that exact question, as you asked it on site #1, isn't going to be a good fit for site #2.
If you're just going to copy the question, you should either leave it where it is (if it has answers) or delete it and ask it on the other site. If you ask two very related questions, each tuned to its site and linked to the other, that's legitimate.