The best bet for any Egyptian link is the Hyksos era. Locations and some details match pretty well, they were semites from the canaan area originally, and when they got booted out a century later, they clearly went somewhere (back to canaan makes sense). Hyksos weren't really any one ethnicity, they were a mix of semitic groups. So it's pretty likely that such groups retreating from Egypt mixed in with the local canaanite population later and might have given rise to some myths and legends surrounding the period.
There are a bunch of parallels to the bible.
For example, the Hyksos capital of Avaris happens to be directly in the center of the "land of goshen" where Joseph supposedly settled the israelites. This could possibly relate to Hyksos leaders inviting in fellow canaanites as additional settlers during the Hyksos period. Such resettlement is well-established historic fact of the period: the Hyksos didn't all come at once, they conquered cities, gradually expanded and a steady flow of settlers from the canaan region joined them.
Also if you read about Josephs actions in Egypt, it amounts to (in black and white unambiguous terms) (1) hoard all the food (2) starve the people to get them to cough up all their gold (3) appopriate all the cattle (4) herd the native population into cities thus depriving them of productive land (5) give the left over land to settlers from your home country. It's plausible because that's precisely how colonialist invasions work in practice. It sounds like a verbatim description of what's happened in Gaza, the Boer War, or the Vietnam War. So, my money is on the fact that there's a grain of truth in exodus, but it's putting a positive spin on the 17th century BC when canaanites invaded Egypt and enslaved everyone, then tried to control them by removing them from the farmland.
Also, I think it's in Genesis 39 "Thus he left all that he had in Joseph's hand, and he did not know what he had except for the bread which he ate." basically, it's saying the Pharaoh lost all control of the country, but still existed as a figurehead. "Puppet emperor under house arrest" is what that sounds like to me (think Pizarro and emperor Atahualpa, who was also seen as a living God, and who's capture thus forced servitude from the natives). The whole thing reads like a "how to subjugate other countries" manual. Perhaps the stories were educational back in a barbaric time of how to invade people. And the last act listed by the Israelites was looting all the gold and silver from the remaining citizens before they fled. Of course this was "magical god-powered looting", like we believe that. So it doesn't count under "thou shalt not steal". lol. But it does sound plausible if an uprising is forcing your people out of the conquered country that the last thing you do is mass looting. It's plausible because that's exactly what would happen in that situation.