It's only 350 miles from Cairo to Jerusalem. I walked 110 miles through the alps in 7 days, so walking flat terrain a straight shot would only take 3 or 4 weeks at a slow pace.
The countries are literally neighbors so if Egyptians did have slave labor, their next door neighbor wouldn't be a stretch.
Cairo to the red sea is only 84 miles, 3 to 5 days walk. Right on the other side of the Red Sea is Mt Sinai region.
If they could not enter Israel, and were not wandering or in the Egyptian area, then that would have left Jordan. The Wadi Rum Desert or maybe even the ancient city of Petra.
Smile! If Egypt was 3-4 weeks walk for an Isrealite to go to Egypt, then it was only 3-4 weeks (probably less) for Egyptian soldiers to go from Egypt to Palestine. And, if you used a chariot, with fresh horses stationed in strategic locations, it would have been a lot less.
As JWs, we never ever thought to question why it was never mentioned that Palestine/Israel was part of the Egyptian empire for long periods of time. This is the start of the ironical religious problem of the Jews. Except in mythology they seldom controlled their land. There were always bigger, stronger Empires with foreign gods conquering their land.
Its not much good having the true god as your national divinity, if the old bugger is too weak to resist the armies of the foreign non-gods, is it?