I had aroud 1000 hours in the L series and I never had such movement.
Not even when transporting firefighters that they use to be moving all the time. You can feel how the move of course but it is not like the kind of "movement" you described above.
Is this movement associated to any kind of noise? How many degrees nose down are we talking?
Do you have any "untied load" on the baggage compartment?
Have you talked about this to any certified mechanic?