I skipped over to my library...
THE SIOUX, by Guy Gibbon says that the Sioux were driven out of the Great Lakes area by the Ojibwa, who were THE power in the North prior to the Seven Year’s War.
It’s a terrific book, by the way. Blackwell publishing produces some very nice, comparatively inexpensive volumes that are stuffed with information. Check ‘em out on Amazon.
Since we (the Sioux) are close linguistic cousins to the the Mandan and the Mandan are better historians, my personal belief is that the Mandan followed a southerly route along the river valleys into the Dakotas whereas the Sioux followed a more northerly route via the Great Lakes.
It is not clear from the traditions of either tribe how long this migration took, but it is generally believed to have been compressed within a century, two centuries at the most.
By following the rivers of a region well suited to agriculture, the Mandan naturally took to agriculture just as the Sioux took to fishing and hunting by following a route through more northerly regions adapted to that lifestyle.
IIRC, I read in the autobiography of Blackhawk that the Sioux lived west of the Mississippi. He said that during the Blackhawk War (1857?), the escaping Sauk and Fox were slaughtered by the Sioux as soon as they made it over to the other side of the river. I think there was a foot note that said they lived in parts of Iowa too.