Switching between sport and eco modes-OK when moving?

There’s no problem doing that. All that is actually happening when you do is your telling the ECU that you want to have a more responsive throttle response. Nothing (effectively) mechanically is happening. Your Owners Manual will explain it in more detail.

 

Never thought that was something to worry about and have swapped quite regularly! For me it is if I'm feeling my oats and want to leap off at an intersection. I go into sport mode at the light, leap off, then while cruising, go back to ECO mode. Given that is about the only time I play with sport mode, I figure I've maybe done 5 miles in sport mode in my 8,000 miles of driving so far. Kinda cool, sport mode for a few seconds leaping off, then back to what the Prius is famous for the rest of the time!

will

 

It's also nice to flip out of Sport while in a parking lot. Eco and Normal loosen up the steering at low speeds. Give it a try sometime to feel the difference.