All framers care about is that you deliver the print flat and protected in a way that gives them some storage flexibility until they can get to it. I use exactly the foamcore system you describe. I have three of those things, and rotate them between the my shop and the framing shop as they become empty.
The normal shipping method for large prints is rolling in a tube. Plenty of posts on this forum about that. 3 or 4" tubes are kinder to the recipient than smaller ones. You first roll the print up on an old 2" or 3" core to maneuver it into the shipping tube without undue scraping. You can theoretically sandwich prints flat between some stick cardboard or plywood for shipping, or in a kind of shallow box, but you will be hit with very big oversize shipping charges disproportionate to the weight, and for the largest pieces you may need to ship freight at costs of several hundred $. So tubes are the only real option for shipping prints. But don't deliver prints locally in tubes, unless you know the buyer will themselves ship it.