In Deathly Hallows, Mundungus Fletcher turns into Harry for the movement to the Burrow. Why would Mad-Eye or anyone else choose Mundungous for the task? He is a liar, a cheat and a thief.
So why?
He may be a liar, a cheat and a thief, but he’s also very loyal to the Order, and just a little scared of Moody.
When Sirius introduces Mundungus, he explains that he owes Dumbledore a favour, which motivates his loyalty to the Order:
“He’s useful,” Sirius muttered. “Knows all the crooks — well, he would, seeing as he’s one himself. But he’s also very loyal to Dumbledore, who helped him out of a tight spot once. It pays to have someone like Dung around, he hears things we don’t.”
Given that this plan requires a large number of warm bodies who can be trusted not to blab, Mundungus seems like a pretty safe bet. (Assuming his loyalty towards Dumbledore extends also to the Order and the general anti-Voldemort movement.)
Throughout Order of the Phoenix, there are scenes where Mundungus is a little wary of Moody’s ever-present eye. Moody makes a point of it during their preparations:
“The pairs will be as follows: Mundungus will be traveling with me, by broom —”
“Why’m I with you?” grunted the Harry nearest the back door.
“Because you’re the one that needs watching,” growled Moody, and sure enough, his magical eye did not waver from Mundungus as he continued, “Arthur and Fred —”
He may be a dodgy piece of work, but fear of Moody seems to keep him in line. At least, enough to stop his criminal background interfering with the plan.
He's a warm body, and one who will likely do his best to flee and hide, given he looks like Harry Potter, so he'll look convincing in his escape.