How to unstuck a low-fee transaction sent from MultiBit?

Question

I've sent a transaction with MultiBit Classic or MultiBit HD, but accidentally included too little fee. What can I do to speed up the transaction's confirmation or to replace it?

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^{This is a complementary question to Why is my transaction not getting confirmed and what can I do about it? which answers the question's broader implications and proposes mitigation mostly focused on Bitcoin Core.}