Language is a question of
learning and
reinforcement and
refreshment and this may come as a surprise to many, but this includes our
mother tongue as well.
A lot of people think, the mother tongue just .... well, just
is, but it isn't.
You learn your mother tongue, and, for most of us, it is constantly being (passively)
reinforced and
refreshed. Even if you don't speak a word, you are constantly seeing (reading, advertising, etc) and hearing (listening to others, to the radio, tv, etc) your mother tongue, so you think it's natural, and that it's always there and that you never forget it.
Wrong !!!I have been living in Germany for over 35 years: the first twenty years were in an almost totally german-speaking environment. I spoke maybe an hour of english in a month, otherwise I read, spoke and heard only german.
The result was, that I forgot a lot of english, and when I visited the UK I was often taken for a foreigner - I just didn't sound quite right, I had to keep searching for words.
Even my (later) MiL thought I "spoke good english for a foreigner"

People who move to foreign countries, or even "foreign" counties have two choices:
- encapsulation or assimilation.
Encapsualtion means you join the "colony" there and carry on talking the way you always did, among people who talk the same as you do. (you don't "go native").
Assimilation means you fit in with the new, and as I did, you gradually forget the "old" because the otherwise constant reinforcement/refreshment isn't there.
So to come back to the original question:
if the emigrants are with a
colony of others from the same region, then they will retain their accents or brogues or dialects - for a while - but the longer they live in the new land, and the more they spread out, then the more likely they are to forget the "old" accents and take on the new.
Add to this the fact that many will deliberatley drop an "old" accent in order to assimilate better into a new country, or a new life, then I guess that in the course of a couple of hundred years, apart from small colonies of who have always stayed together, the individual accents will grow more and more like those of the "host" country/county, with just a few "family" words and phrases being left.
Bob
ps.
After I met my wife (scottish) I started speaking english every day again, and my english improved dramatically (constant refreshment, just from being in an english speaking environment).
Although even now, I still have occasions where I use a word, that suddenly springs to mind, and I think: "that's a good word, I haven't used that one for over thirty years !!!"