Every flight has its W&B calculated (and double-checked by a dispatcher). That is one of the reasons why flights close well prior to departure.

While it is more prevalent on smaller aircraft, I have seen passengers moved from front to rear on larger aircraft as well.

As noted, it is generally --- although not always --- necessary on departure. Once the seatbelt sign is off, ask the FA if she can obtain permission to move back (or ask at the time you are moved).