smoothly. If an actor or director feel a line is not working or want to change it then do it. Just make sure the line change stays the same.
I really like the idea of getting actors involved with watching out for continuity. On a independent budget there is a good chance there will not be a script supervisor on hand and the filmmaker will be stretched thin as it is. Having actors keep track of their own continuity will help avoid errors.
If you’re a seasoned filmmaker you’re much better equipped to deal with improvisational dialogue from actors. First time filmmakers can run into trouble when actors are saying lines different ways. Actors are creative and can come up with awesome lines on the spot, if one of those moments happens embrace it. Make a note to the script a line has changed. What you want to avoid is an actor saying a line a different way every take.