Disclaimer: I work at an Amazon delivery station and briefly worked at a FC prior.
From my experience the managers at my warehouse are very open to moving people around to fill different roles if you request it. This is obviously anecdotal but from my experience people tend to want to stick to one specific job rather than move around.
Fair enough, the logistics of different facilities might allow more leeway in some than others - but in the FC where I work now, they're not very flexible about moving people around unless immediate business needs and stow rates allow it (for indirect roles at least.)
Still, I don't think employees should have to ask at all - Amazon should regularly rotate their FC employees through various roles, even within a workday. Doing so would improve performance, attentiveness and morale across the board, reduce repetitive strain injuries, and give each employee an idea of how various roles interrelate.
I'm in compete agreement with you regarding moving people through different roles. This is actually what I've requested and now have varying duties on my different shifts, it makes the work much more enjoyable to me. With that being said I personally probably wouldn't enjoy being moved to different roles throughout one shift on a regular basis. I've had that happen to me before when we were short on staff and it can be more stressful.
From my experience the managers at my warehouse are very open to moving people around to fill different roles if you request it. This is obviously anecdotal but from my experience people tend to want to stick to one specific job rather than move around.