There are few ways to ahieve your requirement:
1) Create custom drag-drop behavior and override CoerceDraggedItems(), where you could return the state.DraggedAppointments and the additional (related) appointments. More info can be found here:
2) Create custom AppointmentSelectionBehavior and override the GetSelectedAppointments(), where you could return the appointments related to the target parameter (the clicked appointment). This behavior will enable you to always select the related appointments, hence they will always be dragged together.
the Telerik team
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