It would also be logical to carry over the free bonus actions and actions to the next turn
I disagree. As a general rule it would be bad practise, even though I know there are individual examples where it might make sense.
Carry-over would make more sense if it was an action point system rather than an... I dunno what to call it, let's say an "action type" system. But generally the "saving" of actions would interfer with the way DnD is trying to simulate and sequence events happening at the same time. Doing nothing for six seconds do not make you able to do twice as much in the next six seconds
