Aircraft Speed on Missile Launch

[KJ_Lesnick]

I understand that if a missile was launched at Mach 6, it would be moving almost 4,000 miles an hour faster than if it was launched sitting still. My question is, wouldn't the speed after launch fall back to the missiles maximum speed? I've heard claims to say that it stayed (hypothetically) almost 4,000 miles faster across it's whole flight... but missiles are goverened by thrust and drag, and missiles produce virtually the same amount of thrust regardless of speed (technically the nozzles are affected by air-pressure I suppose), and the extra drag would be negated by the same thrust as before and eventually you'd fall back to normal speed right?

 

[Jemiba]

Just my attempt of an explanation:
If you launch an unpowered missile, it would lose momentum from the very first moment, until
forward momentum is nil and it plunges down vertically. So you'll get a graph like A.

If launching a powered missile from a stationary object, it starts with velocity = 0 until reaching
its maximum speed, like in graph B.

For launching a missile from a flying aircraft, you simply have to add both graphs geometrically,
shown in red in C. That means, the advantage of a high launching speed is used up after some time,
leaving just the speed, the missile can achieve on its own. How soon, of course, depends on factors
like drag, density of air, thrust of the missiles engine and launching speed.