: 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.

Good question. I'm sure you're right that there were other Southern routes into Mordor. But I imagine that they were all rather far on the Eastern end. If the troops in question began in Western Harad (or South of there) and wanted to end ultimately at the black gate, taking the Eastern route might have meant going weeks out of their way. It may, consequently, have been an issue of convenience (and arrogance) that drove them this way. In fact, to reach the black gate in time for the big battles, they may have had little choice.

The only problem with this theory that I can see is that I don't imagine that there were any elephants in Western Harad. But even if that's true, I guess they could have been "imports."