I didn’t pick up on the “Aha” thing, lol!
I need to just slightly amend what I wrote earlier. I put “ag bog ar aghaidh” but I meant “ag bogadh ar aghaidh” as it’s the verbal noun of the verb “bog” required after “ag”. I just put “bog ar aghaidh” first to give you the verb, then I realised how he was using it with the verbal noun and I added in the “ag” but forgot to change the verb form. In his south dialect the sound afterwards (a schwa) would merge with the “ar” so it sounds like “boga’ar aghaidh”. Anyway, I guess what is most important is just recognising the phrase itself and the meaning.
“Bog” is a very normal word in Irish to mean “to move” (in this sense, as in moving on, but also to move house etc.) and “ar aghaidh” is an extremely common expression across Irish everywhere you will find it, so there is not going to be anything weird or restricted about such a sentence.
I didn’t grow up surrounded by Irish or anything, so I can’t attest to any specifics, but in as much as it seems to me, I think if anyone needed to express the idea of moving on to a different topic, or changing the subject, this is probably what they would say.
So, just to clarify:
bog (verb) = to move
ar aghaidh = on / onwards
bog ar aghaidh = to move on
ag bogadh ar aghaidh = moving on …
(Edit): sorry I butchered some spellings up originally and just fixed them.