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Placing the adjective "uninvited" after the verb "fly" gives the adjective special emphasis. "Uninvited" modifies the pronoun "they".
In the phrase "they fly uninvited" I lean more towards interpreting uninvited as an adverb, not an adjective, though it could be either. They are uninvited (adj: they lack an invitation), they enter uninvited (adv: they enter without an invitation).
Perhaps you can understand that 'uninvited' as meaning 'being uninvited' or 'although being uninvited'. That past participle has a function similar to 'surrounded' in the sentence ''surrounded by the enemies, he had to fight for his life'' (being surrounded...).