It oesn't seem likely that the Lost Tribes are going to come into play in the period around 2200 BC, since this is even before Abraham went to Canaan. Weren't the Lost Tribes scattered around 800 BC to 600 BC by the Assyrians and the Chaldeans?
What this event tat the end of the Egyptian Old Kingdom might relate to is the rise of the foreign (possibly Semitic) Hyksos kings over Egypt, who are believed by many to have been ruling when Joseph became advisor to Pharoah.
Here's an interesting link to a Ten Tribes article that I'd bookmarked:
Yes, they were captured and moved to the fertile crescent area in waves by the Assyrians over a few years time. The single date 722 BC is commonly used as a marker.
I believe the Hyksos link to be a valid one.
The Lost Tribes link you posted is an interesting and thought provoking one. But in my mind the sudden appearance of MILLIONS of Celts in history at the same place and same time the Isrealites go "over the hill" commands attention. The other isolated little splinter groups that claim to be from the Lost Tribes are miniscule in size compared to the Celts, and don't seem to match up in other ways.