We have moved our embryos twice and had no problems at all. The companies that offer this service are specialists and arrange it all with the outgoing and incoming clinics. We had messages on the day to let us know when they had been picked up and dropped off. I was a little anxious both times, understandably, but all has been fine and we have our wonderful toddler from one of those embryos!
We moved our sperm from one to another and they used a cryopreserved carrier which we had to put a hefty deposit on, but we drove it ourselves and all was good.