Hello Isabelle & welcome to the family!
I'm in regular touch with one of the NZ branch (from the same Beds roots). I sent him our tree some time ago, which covers 1701 to the early 1900's.
Sadly, there has been a lot of erroneous info scattered about the internet re our Armstrong family. The truth is that the link pre the 1680's back to Scotland/Ireland/Borders isn't provable, on paper, for our Beds lot. Also, the Siward-Armstrong progenitor theory is almost certainly myth/legend. Siward's son died with no proven male heir. A number of us have been researching the various theories over 30 years & most hit a brick wall. I've studied the first recorded Armstrong in the country, namely Adam Armstrang (sic) who appears in the Wetheral & Lanercost Cartularies in Cumbria in the early 1200s. The family were of knightly status. They were Verderers - highly important posts in that period.
I believe I know how they came to be in Roxburghshire in the 1300's.
I'm involved in YDNA testing to eventually find the truth. We match to people with Canonbie, Dumfriesshire roots.
Lovely to hear from you.
Cheers, Bob