Are political/ideological reasons for career moves acceptable in academia?

A political or ideological answer, albeit honest, is usually a missed opportunity to make a point about the move being a good academic fit.

Your goal as the candidate in this kind of exchange is not just to answer the questions accurately; it's to answer the questions accurately while also trying to convince the committee that you are the right person for the position. Virtually any question can be used to make a point about how excited you are about the institution you'd like to move to, how much you can contribute given your past experience, etc. A political answer doesn't usually help you "sell" yourself.

For example:

1. Why do you want to study at this university? "Because I'm really interested in the work Professor Twist is doing on basket weaving in extreme underwater conditions."
2. Why do you want to leave your current institution? "Because I am looking to gain expertise in the hot new field of basket weaving in extreme underwater conditions, and your faculty are the best in the world in that area, while my current institution doesn't have anyone working in this field." Or "Because I am really interested in working with Professor Twist, I think that with my background in freshwater basket weaving I could contribute to her research in an exciting way."

So even a completely non-controversial political or ideological reason is not necessarily the best reason to use in answer to this kind of question.

ff524 has covered the question from the angle of a PhD student hire, I will discuss it from the angle of (senior) faculty hires, which I think are slightly different in that regard.

If you have already, as you say, 10 years faculty at an institution you are presumably tenured there. When you then apply elsewhere, an undercurrent of the entire application process will be

"How do we know you really want to accept a potential offer, and not just negotiate with your home university?".

Topical fit as mentioned by ff524 is important, but not really a convincingly strong reason in that regard. A political answer may actually be stronger here, if convincing to the committee. Your Brexit example may, for instance, be convincing enough, as you can link it to reduced funding possibilities and potentially job security problems for your spouse. The Putin example - well, maybe it's helpful if the committee is full of die-hard Putin fans as well, but that one is more iffy as it does not so strongly impact academic life.

Obviously, a political answer that goes against the belief system of the committee is never going to help you. To use a recent loaded example, in most left-leaning universities arguing that you would really like to move to the US because the US, unlike your European home, allows you to buy a semi-automatic weapon with relative ease, is neither a convincing reason for a move nor will it strike brownie points on an emotional level with many faculty members.

So to summarize:

Suppose one is happy at one's institution until something happens that makes it impossible to stay given one's strong ideological beliefs. Is it a good idea to mention these beliefs if they are the main reason for wanting to move?

Yes, if (a) the reason will appear to be a "big thing" to most people, (b) the committee is likely to share your view, and (c) it can be linked to impacting your academic life.

I’ve found that, in general, scientists tend to be quite accepting of political or ideological motives as factors in career decisions. To some extent, this will even be true if they disagree with your politics.

That said, the cases you present obviously differ, and thus will engender different degrees of acceptance: the first case takes a strong, political, affirmative stance about something that doesn’t affect you personally. Whereas the second affects you directly.

To illustrate using a more direct comparison, consider these two statements:

1. I don’t want to work in the US because, as a muslim, I no longer feel welcome.
2. I don’t want to work in the US because I disapprove of the Trump administration.

Most (all?) reasonable people will accept (1) as a valid reason. Many (in Europe) will also accept (2) but it’s unarguably much more divisive (I speak from personal experience): even people who agree with you politically might not agree with such a decision.

Here’s another example: a former colleague turned down a position at a very prestigious institute (in favour of a much less prestigious one) because it would have meant working with stem cells, which she opposed on ethical grounds. I and others strongly disagree with her reasoning but we never had a problem accepting it, and it hasn’t impacted our impression of her as a researcher.