Fantastic - good on them! I've heard of a few instances where illegal immigrants in the UK had suffered exploitation, although the same could be said for many "legal" workers here too!
My father came from the Bahamas - although when he came to the UK, in 1948, it was still a British colony and he had a British passport, so there were no issues about moving here. He came to study, and was planning to return, but then he met my mum...
I mention this because shortly after he arrived, there was a large influx of immigrants from the Caribbean to the UK, a policy that was encouraged by the UK government at the time because there was a demand for cheap labour. There is currently a big row going on in the UK because many of the immigrants who arrived at this time did not receive valid documentation, and are now being denied healthcare in their old age because they are seen as illegal immigrants - despite the fact that they've been working here and paying taxes since the 1960s and 70s.
Some of them have probably been returning regularly to the Caribbean over the years, just as you describe with the Mexican immigrants - but they have still spent years fulfilling the UK's demand for bus drivers, cleaners and hospital staff etc in the productive cities of London and Birmingham.
And the reason they were in countries like Jamaica in the first place is because the British empire needed a lot of slave labour to build up its sugar industry in the 18th and 19th centuries! So it's not surprising that there's a big row about this that has resulted in the resignation of a government minister, when people who worked for the UK for decades and whose ancestors sweated and toiled to build up the British Empire are being labelled as illegal immigrants and denied healthcare.