The Death Eaters' flying (with black smoke trail) is not canonical (except Snape), as they apparate in the books, and the characters never discuss the details of this smoke-trail flying in the movies, so we don't and can't know how exactly they are able to cast spells mid-flight.
Speculation might lead one to imagine that someone in the production team, with power, liked the design choice of black (and white) trails, without really caring enough about the non-canonical (and therefore prone to be logically flawed) nature of it.
It could be that someone liked the sound of Voldemort's flying ability in Deathly Hallows and decided to add it as a standard mode of transport for Death Eaters
"And then Harry saw him. Voldemort was flying like smoke on the wind, without broomstick or thestral to hold him, his snake-like face gleaming out of the blackness, his white fingers raising his wand again"
—Harry Potter witnesses Lord Voldemort flying unsupported
Though, in the movies, we see the effect for the first time debatably in Goblet of Fire, when the Death Eaters fly down from the Dark Mark at the graveyard scene, and otherwise in Order of the Phoenix, both of which happen before the canonical reference to Voldemort's flight, so that's unlikely.
It seems that Snape (Death Eater) can fly the same way Voldemort does, in the books. It is mentioned in Deathly Hallows.
When Harry looked up again, Snape was in
full flight, McGonagall, Flitwick, and Sprout all thundering after
“He jumped,” said Professor McGonagall as Harry and Luna ran
into the room
“You mean he’s dead?” Harry sprinted to the window [...]
“No, he’s not dead,” said McGonagall bitterly. “[...] he seems to have learned
a few tricks from his master.”
—Harry Potter and the Deathly Hallows, Chapter 30, The Sacking of Severus Snape, J.K Rowling
I interpret "Snape was in full flight" to be a metaphorical reference to "running away at full speed", rather than literally flying, but the rest seems to indicate that he, like Voldemort, can fly (presumably with the smoke effects described in the quote above).
These two instances seem to be the only references to this kind of transportation for Death Eaters/Voldemort, and provide no insight as to how exactly it works and allows them to cast spells mid-flight, nor is it ever implied that any other Death Eaters we see have this ability (in fact, the surprise at Voldemort's flight ability, and McGonagall's comment about Snape imply the opposite), so my answer remains reliable, valid and intact.