You buy one of those remote control toy helicopters. You bring it into an elevator. The elevator goes up. Does the helicopter hit the floor or does the floor of the elevator push the air up into the bottom of the helicopter so that it maintains altitude relative to the elevator? The same question can be reversed. If the elevator goes down will it hit the cieling? If the elevator is air-tight, would it make a difference?
The air in an elevator does tend to move with the elevator, because it has relatively little inertia. However, thinking about the problem in these terms seems, to me, misleading. The simplest way to think about this is to consider the acceleration of the elevator as and addition to the normal acceleration due to gravity.
In this light, it would be as if the helicopter were momentarily heavier wen the elevator accelerated upwards, and momentarily lighter when it accelerates downwards. This would inevitably cause changes in the height of the helicopter above the floor of the elevator, but I expect that most real-world elevators would not accelerate fast enough nor long enough for the helicopter to be smashed to the floor.
Of course, toy helicopters are not all alike, so your mileage may vary!
I'm going to try to answer this as empirically as possible. Firstly, does the air move relative to the walls as a result of the elevator motion? I believe the answer is "no", but I need to substantiate this. There are two fundamental phenomena we should be concerned with:
I will continue to refer to the acceleration as an additional force, and in fact, an adjustment to gravity. The elevator physics behave as if gravity suddenly changed, and this is completely valid because there is a solid non-inertial reference frame which is the walls of the elevator.
Both of the aforementioned factors are negligible. For fast elevators, we may be discussing an acceleration 20% that of normal gravity, and if gravity isn't significant for these factors then the added acceleration won't be either. Indeed, the pressure and density difference from the top to the bottom of the elevator is easily taken to be negligible and so is the wind in elevators due to temperature gradients. I think we've sufficiently addressed this point - that the movement won't affect the air. The air physics are dictated by fluid mechanics and both gravity and the additional acceleration can be neglected for the confines of the elevator. This is obviously not the case of the actual helicopter.
The important observation is that before the elevator starts moving the helicopter is in a balanced state. There are only two forces acting on the helicopter so this problem is relative easy. There is the upward fluid force from the air and there is the downward gravity force from gravity. Now we adjust the pseudo-acceleration due to the combination of [gravity + acceleration]. When this pseudo net gravity acceleration changes the helicopter will accelerate. I need to make a certain claim abundantly clear:
You have 2 balanced forces. One changes, the other stays the same. Thus, the craft accelerates. If the elevator starts moving upward, the helicopter moves down, and visa-versa. This is pretty much incontrovertible. There is no physical argument for the fluid force to change at the original hover point. I need to add one adjustment, which is that the geometry of the elevator box does affect the fluid force, and this is a complicated phenomenon that would need computational fluid calculations to assess correctly. Helicopter pilots know that, everything else constant, the lift when you are close to ground is different from when you are high in the air. Could this keep it from hitting the floor when the elevator moves up? It is possible, and in this case there would be some new equilibrium location (lower than the original location) where the lift is sufficient to overcome gravity + the upward acceleration of the elevator. In most cases, however, it would probably just hit the ground.
Lift of any aircraft is relative to the air mass it is in.
If the air mass temporarily accelerates upward, such as for 1 second at 0.1g, then if the aircraft were to maintain its position in that air mass, it would have to generate an additional 0.1g of upward lift, during that second.
If it is not controlled to generate that extra lift, then yes, it will fall gently toward the floor during that 1 second.
It will not simply stay at a constant height above the earth because the relative upward velocity of the air will, by itself, generate extra lift.
(An extreme example of this is when a real aircraft happens to wander into the up and down drafts inside a thunderstorm. The g-forces can easily "void the warranty".)
Assuming an initially vertically stationary hover I would say that the floor closes in on the helicopter to some small distance and the helicopter then moves up with the floor to some degree.
My reasoning is that the helicopter is creating a downwards force via its rotors but this is poorly coupled to the elevator floor and so is essentially independent of its movement. Until the floor closes in on some length where the downward force couples with the floor of the elevator and allows the helicopter to move upwards at some velocity. This velocity is very hard to estimate without a hatful of assumptions.