What does Luke 4:31 mean?

Luke frequently begins segments of his gospel with a Greek word most often translated as "and." This doesn't necessarily imply the same thing as words such as "then," or "afterward." It can mean something happening later in time, but in many instances, it does not. That means when Luke introduces a new segment of this gospel with the word "and," it shouldn't be seen as a direct statement about the order of events. This helps explain why similar events in the four Gospels are not in identical order in all books.

In this case, the last passage described Jesus' disastrous visit to His childhood hometown (Luke 4:16). That ended with a mob trying to throw Him off a cliff (Luke 4:28–30). Whether this section describes where Jesus went from there, or gives more detail about His preaching prior to His return to Nazareth, is not clear.

Jesus' pattern during this phase of His earthly ministry is to participate in Sabbath meetings in the local synagogue, teaching and preaching, as well as performing acts of healing.