Two were annoying about Frenchmen having moved in (10)

I believe the answer is:

immigrated

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'moved in' is the definition.
The answer and definition can be both related to change as well as being past participle verbs.
Maybe they are linked in a way I don't understand?

'two were annoying about frenchmen' is the wordplay.
'two' becomes 'II' (Roman numerals).
'were annoying' becomes 'grated' (I can't explain this - if you can you should believe this answer much more).
'about' means one lot of letters goes inside another.
'frenchmen' becomes 'MM' (two M's - M is short for Monsieur).
'ii'+'grated'='iigrated'
'iigrated' placed around 'mm' is 'IMMIGRATED'.

'having' is the link.

Can you help me to learn more?

I've seen this clue in The Times.

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