Apologies, I have not read the books in a long time. I was curious how John Carter travelled to Mars? My initial thought was that it was some "new age" (for the time?) Astral Projection type process. His body itself stayed on earth, but he manifest his Astral form on Mars?
As I think a bit more about War of the Worlds: Offensive how would the Humans get to Mars? I can think of only a few examples and those would be Space: 1889, Flash Gordon, and John Carter. The Space: 1889 and Flash Gordon models seem less likely in the AQMF timeframe, so that left me to recall the John Carter method of travel.
The Space 1889 world depended on the ‘luminiferous ether” which permeated all matter. By clever invention, it could be harnessed to provide propulsion in vacuum by the “ether propeller.” Ether propellers were invented by famous electromagnetic experimenters like Edison, Tesla, Armstrong and others.
Ether propellers did not work at low altitudes due to atmospheric interference, so they could only operate at high altitudes (above 12.000ft). However, when “liftwood” was discovered on Mars the problem of low altitude flight was overcome. Liftwood created a gravity shadow above it rendering things above it weightless. This enabled a combined liftwood/ether propeller craft to be created for surface take-off and vacuum flight. Before lift wood was available, ether propeller flight had to be from high mountaintops or from balloons or dirigibles. The laboratories working on them were located far up mountains.
While we are at it, let us remember “Cavorite,” a synthetic analog of lift wood. HG Wells created it his book The First Men In the Moon. By blocking gravity on one side or the other, his spacecraft could liftoff and maneuver.