I've seen them repair over time and get back up to 9kts (or faster to regroup with the convoy.)
I've also seen them lose their engines/props and stay put until the DD gives up on me. At which point I salute them with my deck gun for being so courteous.
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"I wish to have no connection with any ship that does not sail fast. For I intend to go in harm's way." - Captain John Paul Jones, 16 November 1778, in a letter to le Ray de Chaumont.