I've been trying a slightly different approach and I'm having some luck so far. I can't move all the databases at once but I can move one at a time. Here's the basic process so far.
1) Login to the old machine with your data and dump the database with a few extra flags...these will iron out some of the incompatibilities between sql versions:
mysqldump -uroot -p --quote-names --allow-keywords db_name > db_name.sql
2) While still logged ino to old machine, use scp to securely copy the file to the new machine. Assuming the new machine's ip is 11.11.11.11 type this:
scp -Cp db_name.sql root@11.11.11.11:/var/lib/mysql/
NOTE: the directory /var/lib/mysql must exist on your new machine. if it doesn't, you can choose some other directory that does exist. the trailing backslash is really important.
3) Log in to the new machine and cd to the dir where you copied the file in the last step:
cd /var/lib/mysql
4) On the new machine, login to mysql, create the database using the same name as your original database name in step 1, and then quit
mysql -uroot -p
create database db_name
quit
5) import the database to myslq on the new machine:
mysql -uroot -p db_name < db_name.sql
That appears to be working for me now. If you don't have the root password this will probably be trickier.