muting during chord progression

I'm doing an exercise (chord progression) which includes some slash chords. One of the chord changes is from C/G to F (E-based barre chord using all 6 strings...not the four string version).

My problem is that all chord changes are smooth in the sense that as I change from one chord to the next, the previous chord (or at least some of the strings from it) continues to ring a bit as I hit the second one...so there is some flow/continuity to the sound.

Except: as I go from C/G to F, as soon as I release the C/G chord, all its sound instantly stops, so when I hit the F chord a second later, it's a jerky stop/start sound.

Am I doing anything wrong...is there any way to keep the C/G chord ringing a bit as I hit the next chord? Thanks...Patrick