Do people want to work in overseas branches? As an example let's take the University of Southampton, Malaysia branch. At face level, the university opened a branch in Malaysia because it's a symbiotic relationship: the university earns more revenue + spread its brand name, and Malaysia gets to leverage the university's expertise to train its future engineers + provide employment (administrators, academics, support staff, etc).

For the Malaysia branch to have some legitimacy, some of its staff must be from the main UK campus. From the branch's "our lecturers" page, this is indeed the case.

My question is, did these lecturers volunteer to go to Malaysia, or were they assigned by the university? Is it a good thing to be able to say one has worked in an overseas branch, or is it a distraction that must be endured for the sake of the university?
You will have to ask these specific people to know the exact details. But in general, in the UK, the employer cannot force you to relocate unless you have a mobility clause in your contract and the relocation is reasonable. It's unlikely that the employees had a relocation clause to Malaysia right from the start, and asking an employee to relocate halfway around the world is probably not reasonable. So no, these people certainly volunteered to move abroad.

More generally, most (tenured) academics expect to be considered more like citizens of their universities than the usual work-related subordination relationship implies, and tend to want to have some agency in defining their working conditions and goals. In particular, this implies stable employment conditions unless the employee wants to change.

As for why they wanted to move abroad: there can be plenty of reasons. They want to form new collaborations with people from there. They want to strengthen existing collaborations with people they already work with. They want to take up the challenge of teaching to students in another country. Their pay is better compared to the local standards of living. They want to learn a new language. They want to discover a new country. Etc.

(PS: This isn't the 16th century anymore. The mindset of thinking "We'll open up an establishment in the colonies to spread our wealth and knowledge for the indigenous people" should be a thing of the past. There are simply smart people living and working in Malaysia, and the university of Southampton wants to work with these people.)