Watch the Not Always Easy online guitar lesson by Brad Carlton from Chord Studies: Slash Chords Vol. 2
This progression is written around the bass line. Starting with an F / C, your third and fourth fingers will be guide fingers and they move up a whole step to get G / B. Your fourth finger will be a pivot when you go to the B♭m6 and then act as a guide as you move down the B♭m 6/9. Remember: good fingerings aren't always the most comfortable, especially early on in your playing.