I didn't find any indication in the program or the SOLIDWORKS KB that would indicate any ability to create a thermal analysis with a Heat Power vs Position. About the easiest way I could think to get close was to create several discrete faces along the path you want the Heat Power to move. Then in the thermal study you would setup several Heat Power definitions that only turn on for a fraction of the total time and apply them in sequence to the discrete faces on the path.
The faces were each setup to have a 5 watt power applied to them. They use a time curve to start at 0 watts, step up to 5 watts for 1 second, then step back down to 0 watts. It is definitely not a perfect representation of what you describe but it is as close as I can think to make right now.
Thank you. That's a really good idea for a work around. I was trying to figure out how to do it this way but I didn't think to make each square change heat power over time in a cascade.