From
JED database:
Quote:
The BMD-3 can be airdropped from transport aircraft with the complete crew remaining inside the vehicle. In the past, the crew was dropped separately, and it often took a considerable time for them to locate the vehicle. By dropping the vehicle with its crew ready in the vehicle, the element of surprise associated with airborne operations is enhanced.
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This confirms what you said about early BMDs, but adds its own grain to the problem, since it means some vehicles could be dropped ready, in some extent. This text sounds like manufacturer advertisement, so I won't trust it too much regarding the ability of this thing to roll on without any packaging to remove or parts to mount back on...