How much is covered by the donation recipient varies by state. In California (one of the two states I know something about), you donate directly to the School of Medicine (or whatever). In my aunt’s case, they picked up her body. In my mother’s case (Illinois) you have to donate to a clearinghouse, and the family pays for transport, which is not cheap – it’s not like you can just arrange an Uber or even transport it yourself.
In both cases, the recipient paid for cremation. The donating familyhas the option of getting the ashes.