I have used The Foam Book's technique of using polyfoam glued into tubes, and glued to the mouthplate, then covered with fabric.
Often though I won't use tubes but instead make a "finger pocket" that all my fingers go into. This can be especially helpful if the puppet might be used by a "leftie".
Another of my puppets uses a jersey work glove glued to the mouthplate. It takes a few seconds to put him on, but once he's on his responsiveness is lighning-quick!
Like so many things in puppetry, there are no "approved" ways. At the end of the day, it's whatever works for you!