Changing departure times

This happened to a relative: His flight was to leave at hour X, then the airline change it to X+4 hours, and then moved it back to X+20 minutes. He received a text from the airline about the final departure time, after he was on board and the flight was away from the gate. You can imagine the number of people who missed the flight because they thought it would be much later.

Is this common? Does the airline take responsibility?

It’s not unheard of, no.
It is the passengers responsibility to be at the gate for an on time departure even if the flight is late. Texts are nice but as your relative found out things can change with very little notice so “nice to have features” are not always right up to date.