Just providing a cross-reference here. I asked for clarification on this issue, but ended up getting the answers through experimentation.

If one creates a stored procedure that always returns a known number of result sets (probably good practice to do this anyway), the only difference to the "traditional" usage of DBI is that you need to call $sth->more_results() before retrieving each new set (otherwise you don't get the new set.)

So, this means that $sth->more_results() doesn't just mean "are there more results?" but is also the command to retrieve the next result set.

Please see the node I have referenced above for examples.