If I move a planting, do its tasks move too?

Yes, so a late spring never means rescheduling a dozen tasks by hand. Tasks are tied to their planting: drag it a week later and every task not yet done moves a week later, even one you had rescheduled by hand.

Tasks already done stay where they were. Once a step such as the transplanting is done, its date is the day you completed it and no longer moves, but you can still move the steps that follow.

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