How do I use Canon speedlights off-camera with a Sony A7 III?

The only triggering system where you can use legacy OEM TTL speedlights cross-brand is the Cactus V6II X-TTL system. However, Cactus may be out of the photographic gear business. And the entire system consists of the transceivers and two single-pin speedlights that cannot do TTL/HSS on any camera hotshoe; they can only do TTL/HSS as off-camera radio slaves. There are no larger lights in the system.

Godox does not allow you to use legacy OEM TTL speedlights cross-brand with TTL and HSS. The X1R receiver is dedicated to a single brand, and only delivers TTL/HSS with a same-brand camera, transmitter, and flash.

So, for example, you can only get TTL/HSS with a Godox XPro-C on a Canon camera hotshoe, with a 580EX II on an X1R-C. Trying to use an XPro-S on a Sony camera hotshoe with the X1R-C would be manual communication. You would definitely have sync, and you'd probably have group on/off control, but remote power control, TTL, and HSS are improbable (I've seen various conflicting reports of what function you would have).

The X1R-S would be if you were using a Sony HVL speedlight with an XPro-S on a Sony camera hotshoe.

What is cross-brand in the Godox system are the transceivers or receivers built into Godox strobes. So, if, for example, you had gotten a Godox TT685-C as your Canon speedlight, that could be used as a TTL/HSS radio slave to an XPro-S on your Sony camera. I use a TT685-C that way with an XPro-C on my Canon 5DMkII, an XPro-F on my Fuji X100T, and an XPro-O on my Panasonic GX7:

Whatever goes directly on the hotshoe of the camera (transmitter, speedlight) has to match the brand of the camera, though.

Generally with Godox, it's a matter of swapping your legacy flashes for Godox lights. It may be best to sell your Canon flash gear, and get Godox gear for Sony instead.