The one bit I'd change is I'd plunge the meat, in bag, into an ice bath to stop the cook and cool the meat very quickly (getting it out of the danger zone as fast as possible).
Just having it surrounded by ice isn't as effective as it is in ice water (and the surrounding ice would melt, producing air gaps, etc). Depending on the thickness of the meat, you may want to leave it in the ice bath up to an hour. (take about half the time from the thickness charts for cooking).
I wouldn't transport it in hot water - unlikely your food would cool enough to become dangerous, but it's less than ideal - keep in mind you have to get your bath ready to receive the meat on the other side.
Lastly...do you not like these relatives? Seems almost cruel to cook meat in their home, then amskray with it!!