Hard drive moved from Firewire to USB3 doesn't read

Got a new iMac so I took my external back up drive out of it's Firewire enclosure and moved it into a new USB3 enclosure so that I could easily migrate from the back up using speedy USB3 (since there is no Firewire on the new iMac). When I plugged the drive in, the system said that it was not recognised and would I like to reformat? Well, no, I don't, so I put it back into the old enclosure and plugged it in via USB2 and there was, all safe. So is it normal that the disk can no longer be read when it is moved into a USB3 enclosure? If so, I guess I have to back up my other external drive before I move it into the new USB3 enclosure, but I wanted to make sure this was correct behaviour. Any thoughts on this would be appreciated.