Why would you think that an air gap would act as a
barrier to heat transfer, in "cavity walls", or anywhere else?
As we all (should) know
"Heat can travel from the hotter to the colder - by Conduction, Convection and Radiation
and that's a physical law"
(Shades of Flanders and Swan.
and
https://mathshistory.st-andrews.ac.uk/Extras/Thermodynamics/ )
In any "air gap", heat can travel by both Radiation and, more importantly, by Convection - with Conduction being minimal.
However, while the Radiation
might not be very significant,
in a large "air gap" Convection certainly
will be significant!
The (warmer) "Convection" air currents will (eventually) come in contact with a colder surface and transfer the heat in them to it.
They will then
sink down, to be
re-warmed and repeat the cycle!
That is why "insulation"
is installed in "Cavity Walls".
The main function of this "insulation" is to reduce, as far as reasonably possible,
heat-transfer - by means of "Convection".
It also reduces heat-transfer by means of Radiation.