This may seem to be drifting from the point, but not really as it is still very firmly connected to the family at the top of the thread.
I am pretty sure I have confirmed the "gateway" person linking the Beakhust(r)st families in UK with Ireland.
Albert Edward Beakhust was born St Pancras London 1884, son of Joseph William Beakhust and Eva Turton. Joseph William was in turn son of Thomas William Beakhust and Eliza Verge.
Albert Edward joined the Middlesex regiment in london in 1903, and within weeks was posted to Dublin. Also within weeks he was discharged "unlikely to become an effective soldier" or words to that effect but did not specify why. He remained in Dublin. In 1911 census he was living in a church run soldiers home. He married Ellen Orr in 1914, and later in 1914 signed up again to the Middlesex. His birthplace is given as "St Pancras, Dublin" which is unlikely as I cannot find a St Pancras there. All the other details match, though, so only the "Dublin" is incorrect, and may have been an assumption by the recruiting officer. Again within weeks he was discharged for the same reason as before. A pensions department slip dated 1920 gives the reason as "deaf". I imagine he joined in 1914 because of the very considerable public pressure put on able bodied men who did not. Perhaps it was more simple to join and get a discharge paper than to keep having to explain.
He had children Albert, George, Ellen and Gerald (may have been more, perhaps?).
Surname spellings seem to have migrated to beakhuRst in Ireland but throughout history you find even if the record says one, an index or transcript will say the other!
Sorry, lots of time has passed since my last post, but hope this is useful.
I have so far not found a Beakhust or Beakhurst outside of England, who (unless adopted) is not descended from Thomas William Beakhust (1813), or his brother Bernard (1819). Within England I haven't nailed down the family in Kent.