I use clean, white (although color doesn't really matter at all, just my preference) wash cloths for 1/25 models, hand towels for larger. I place the model diagonally (pointing toward one of the corners) the center of the wash cloth. Then I lift it by the two corners alongside the length of the model and place it washcloth and all into an inverted AMT clear plastic display case. Roll the corners alongside, in front, and behind the model to take up any slack in the case, then put the base on it (upside down) as a cover. I either use a sufficient rubber band or masking tape to secure the top (bottom) of the case.
I can transport one model this way, or if I'm taking multiple models somewhere, I have some corrugated cardboard cartons that will hold three inverted display cases. In that instance, I load the cases into the carton, in towels, and tape the top of the carton shut. No damages, and everything is re-useable. I keep a supply of display cases, wash cloths, hand towels and the cardboard cartons dedicated to just transporting models.
For large scale models, I have some larger plastic cases and cartons and use hand towels, tea towels, or small bath towels as appropriate.