To move backlogs from one section to another section on the Backlog Board, one needs to have the Edit work items in this node permission for the Area and Iteration path. You can deny the permission to disable the ability for that specific engineer.
Go to the team project admin page (https://vsoaccount.visualstudio.com/DefaultCollection/teamprojectname/_admin/_areas), right click the Area and select Security. Select the engineer you would like to set the permission to set disable that permission. See:

Per my above screenshot, user Victory Song can't move work items which is under Agile area. (and she can't edit work items under Agile area either.)