This is probably the most complex phenomenon in drum physics. it is know as “Fluid / Structure interaction” and is often studied in fluid dynamics and sometime in NVH engineering (Aerospace and automotive typically)
This has to be considered whenever a vibrating surface generating sound waves is “back loaded” with the “response” of the environment, therefore impeding its motion. In the case of drums, studies on timpani showed that this phenomenon is really a first order one as it does shift the membrane modes from a “theroetical non harmonic” series to a nearly harmonic one for the first modes (or partials).
As the main sound generation mechanism are the vibrating membrane, any phenomenon significantly modifying the vibration has to be considered – drummers know this perfectly as they would sometimes carefully tension the heads to even tension, and would take tremendous care in the bearing edge geometry and surface condition.
A real membrane is vibrating in a “sea of air” as described by Rossing (I strongly recommend the reading of Thomas Rossing’s “Science of Percussion instruments” ISBN-13 978-981-02-4158-2), as a result and depending on the considered mode shape and cavity shape, some modes are “mass loaded” and shift down in frequency, some other are “stiffened” by the air compression and tend to shift upwards.
Repercussion studied this effect on the snare drum and noted the following difference between a theoretical membrane (that is, vibrating in vacuo with no bending stiffness) and measurement on a 14″ snare.
As frequency (and mode shape complexity) increases, the difference increases too : the real membrane shows significantly higher natural frequencies than the theory.
The modes with nodal diameter (lines of no motion across the heads) tend to be lowered from air loading for the first 2 modes (1,1) and (2,1) but are the “fastest” increase as frequency goes up (Air compression effect and membrane flexibility)
Modes with nodal circles (circles of no motion on the head) seem to be systematically shifted up (stiffness effect from the air compression) as frequency increases.
Bottomline : a snare drum will see some significant effect of air loading, probably less than a Timpani because of cavity geometry (the one of a timpani being closed) but still some influence on head vibration and the produced tone.
From there, it is easy to understand that the venting system construction will have a significant effect on the behaviour of this “sea of air” and therefore on the sound produced. The quicker / easier sound waves flow out of the drum, the more effect will be seen on the drumhead vibration.