So named after Mike DeSisto's famous "Pass Thru...Move On!" choreography, in which squares of dancers are arranged and rearranged in a huge matrix.
Typically, the caller starts out with "normal" squares, and has heads or sides perform a Square Thru 4, followed by one or more calls which convert the square into facing lines.
At that point, all squares are directed to "pass through", and then "move on!" until they come face to face with another line of dancers.
Also known as "Progressive Squares".
If a line of dancers is facing a wall (e.g., not another line of dancers), their job is to perform a California Twirl or Partner Trade to face back into the set, at which point they're facing another line of dancers who have just moved on from a different square.
The caller moves the dancers around for a while, forms them back into lines, and continues the above process. In this fashion, an entire square can be moved from front to back or side to side on the dance floor.
Some convention committees choose to offer this as a specialty tip. Other committees have been known to incorporate Move-On into opening or closing ceremonies and/or Trail In/Trail's End dances.
YouTube videos from DC Diamond Circulate 2009