: : : I imagine the troops must be coming from the lands down South.

: : : Why then are they marching across Ithilien to reach Mordor.
: : : Surely there must be an entrance to Mordor in the South or
: : : South-East. Or else how would Sauron have moved his troops
: : : before gaining control over Ithilien.

: : There was a Nargil pass farther south that was dropped from the maps and story, but it may still have been there in Tolkien's conception. Sauron could have insisted on sending troops through Ithilien as a means of asserting his presence there.

: It is also possible that the well-paved road running northward through Ithilien was faster than entering the southern Ephel Duath, marching across the plains of Nurn, climbing the plateau of Gorgoroth and traveling across its barren and broken landscape until arriving at their destination (especially if their destination was one of the northern strongholds - the Isenmouthe, Durthang or the Morannon - to begin with). In much the same way, it would have been easier/faster for Sauron's Easterling allies to approach the Morannon via the flat lands north of the Ered Lithui (the same route taken by the Wainriders and the Men of Khand in TA 1944) then to enter Mordor through the eastern gap between the mountain ranges and march across the hostile landscape.

Food / Fodder requirements aren't really a concern for a well supplied army, that's what the supply train is for.