The correct answer grammatically is to say "their working". The reason is that "working" in this sentence is a gerund. A gerund is the present participle form of a verb used as a noun (the present participle is the "-ing" form of a verb like "going" or "running" for example). When used as a gerund, it is being used in the same way as a noun and is handled grammatically just like a noun . If you took "working" out of the sentence above and replaced it with a word that is obviously a noun, you would have no problem seeing the right grammar. For example if we replaced "working" with a simple noun like "house", the sentence would read "One trend is the move away from their house ..." and you would easily see that the alternative ""One trend is the move away from them house ..." is obviously incorrect. Since "going" is a gerund, it needs to be treated just like a noun (like "house") and so the sentence that uses "going" should also use "their" just like the sentence that uses "house". Having said that, though, I should point out that many native English speakers are woefully ignorant about the use of the gerund and it is very common to hear native speakers say "them working" in a sentence like your example. It's not grammatically correct but it is so commonly heard that, sadly, I don't think most people even notice the error, and listeners would certainly still know what you mean..

Gareth's answer is correct and very thorough. It's true (as both Gareth and Chris have noted) that in everyday conversation, many people don't use the correct form (and don't understand it). However, I would say that in writing, especially professional writing, the correct form (possessive + gerund) is still preferred. To me, "them working" sounds wrong, or at least very informal. I would never write it this way, and I probably wouldn't say it this way when speaking, either, although many other people do. I actually think Jose's solution is the best one. You can easily just say "the move away from working for the same employer for years," and your meaning is perfectly clear. It's always good to avoid problematic grammar points when possible!

One trend is the move away from it. The noun seems to be equivalent to "they are working for ...." not "their cat" While Gareth's answer seems likely correct, the reasoning is a little difficult to follow, for me. . For me his explanation would directly cover it if it were their something, as in a possessive. but here, it seems "they" are part of the noun-equivalent. . Not questioning the answer, I assume that answer is indeed the grammar rule, just whether the explanation as given is valid reasoning.

This is a complex topic and there are multiple acceptable answers. This was discussed in detail a day ago. https://www.italki.com/question/458701

With this particular example, I think the correct and usual way to say this would be: "One trend is the move away from working for the same employer for years." I think using any of the variations "they", "their", or "them" doesn't work here.