I'm working on the first of Anders' fingerpicking lessons, and found that Casey's Last Ride (Kris Kristoffersen, though many of you will not be surprised that I think JD did the definitive version) is an ideal song on which to practice this first pattern.
The original key has all of the chords that Anders suggests practicing with, including the Dm, Am and F. No barre chords. All but one of the changes happen right on the first beat of the measure, though there is one split measure which is good practice. The basic pattern that Anders teaches fits the rhythm of the song perfectly. It's a nice even slow tempo. And it's a gorgeous and beautifully written song to add to your repetoire as a bonus.
If someone had sat down to write the perfect basic fingerpicking practice song, I don't think they could have done better.
Kris' version (key of C):https://www.youtube.com/watch?v=JIBq1Jkm7YU
JD's version (key of G):https://www.youtube.com/watch?v=nsL-kkmpC0o
"You can get what you want or you can just get old." Billy Joel