At the time most of these people emigrated (the 1700's and early 1800's) there wasn't really a singular "English" identity
Yes there was. In fact there was much more of one than a French identity in France, a German identity in the German states, etc.
Of course Englishness differed (and differs) in different parts of England, but it is a big country (population-wise at least) and that kind of thing is absolutely normal. Of course it can also be true of very small countries, as in Wales.
No, they most certainly did regard themselves as English and did so every bit as much as people further south. The Venerable Bede was a Northumbrian you know.
The Border was (and is) lightly populated and there isn't much in the way of settlement continuity around it (Berwick-upon-Tweed is like the only significant exception). You cross the border and the accents change pretty much automatically. Quite different from the Anglo-Welsh border which is 'deep' and very porous. Cumberland is very English, Dumfriesshire is very Scottish.
There was clearly a lot of migration from Cumberland, and many families with branches on both sides of the English-Scottish border (Taylor, Graham, Jackson, etc). Wasn't the relative emptiness you speak of in terms of the border due to a large degree to the actions of the Crown to pacify the region in the 17th and 18th century?
