Additional features, settings, or issues not covered by specific Microsoft Teams categories
Is it possible for the owner or other users to be notified by email when a task is moved by an assigned person?
No, I am afraid it's not feasible at present as moving a task to another bucket is not an activity that would trigger a notification currently.
Given this, I tried submitting this idea to the dedicated feedback portal for Microsoft Planner. I'll leave the link below so that you and others who see our discussion here can vote or comment there as well. Hopefully this feature can be added in the future.
Email notification when a task is moved by an assigned person
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