Best way to work with data from multiple MySQL databases?

Hi. I'm looking for a bit of advice. I'm a bit of a newbie when it
comes to MySQL setup and administration; I'm a software developer,
I've mostly used it as a backend.

My client has a server with a number of Moodle instances on it - each
has its' own database, one for each client of theirs. They would like
to do some reporting and tracking across all instances - and there are
interesting scenarios to look out for, such as "merging" people's
results when the same user appears in more than one location.

The reporting program will also have a database of its' own, with some
additional tables.

I'm wondering what the least painful way might be to work with the
data from all of these individual databases, plus the application's
database, which will all be sitting on the same server.

Any suggestions?

Remi.