Architects and builders have the ability to move a grid to a new folder.

To move a grid to a new folder you need to be in edit mode (pencil action button), then you simply drag-and-drop (left mouse click hold on the crosshairs button) the grid to the new folder location (via the folder pane on the left-hand side of the grid library page). 

You will need to expand the folder view as required in order to see the recipient folder you wish to move the grid into. 

For professional and enterprise level plans, the ability to move grids to folders may be restricted as a function of configurable access controls applied. If access controls have been applied to your account, you can only move a grid to a new folder if you have "edit" rights for the grid and "edit" rights for the folder you are seeking to move the grid into. If you do not have "edit" rights for the folder you are seeking to drag/drop a grid into, you will be presented with the not-permitted (no go sign).

See "What is configurable access control?".