Cage deformations allow you to use a low-resolution geometry to drive the deformation of a high-resolution object. They are similar to arbitrarily shaped lattices.
Each point on the deformed object is weighted to the points or polygons on the cage, similarly to how an envelope's points are weighted to its deformers. However, you cannot paint the weight assignments nor adjust them manually. You can also have multiple cage deformations on the same object, unlike envelopes.
Cage Deformations and Construction Modes
Cage deformations use a "rest" shape of the cage object to determine how to modify the deformed object, as well as how points on the deformed object are weighted to the points or polygons of the cage. Any deformations you apply to the cage on top of this rest shape affect the deformed object as well.
When you apply a cage deformation, the rest shape is defined by the current top of the modeling construction region in the cage object's geometry operator stack. The cage-deformed object is affected by any new deformations above this point in the modeling region, as well as in the other regions. If the cage object already has deformations above the modeling region, the shape of the cage-deformed object changes as soon as you apply the cage deformation.
The rest shape can be affected by modifications you make to the cage object. If you freeze all or part of the cage object's modeling region and "bake" the position in the stack where the deformation reads the rest shape, then the bottom of the stack becomes the rest shape.
When you set the Construction Mode Display of 3D views to Sync with current mode, you may or may not see the effect of the cage deformation depending on which region of the cage-deformed object contains the Cage Deform Operator. However, if the effect is visible, then the deformed object is always affected by all operators on the cage object no matter what region they are in.
For more information about construction modes in general, see Construction Modes and Regions.
Select the objects, branches, groups, models, clusters, or points to be deformed.
Choose Modify 
Deform by Cage from the Model toolbar. This command is also available on the Animate and Render toolbars.
Pick one or more objects, branches, or groups to act as cage deformers.
Note that you cannot use point cloud objects as cage deformers.
When you have finished picking deformers, right-click in a 3D view to finish. The Automatic Cage Assignment Property Editor [Properties Reference] opens.
Set the parameters as desired. These parameters control how the points on the deformed object are weighted to points or polygons on the cage.
When you pick an object as a cage deformer, its Render Visibility is turned off in its Visibility property and its display mode is set to Wireframe in its Display properties.
To see the effect of the display settings in non-wireframe views, make sure that Override Object Properties is off in the Display Mode menu. See Choosing How to Display Specific Objects [Viewing and Playback].
Cage deformations can involve a large amount of data. Here are some tips for making them as fast as possible:
Make sure that your cage has the fewest points or polygons necessary for the effect you want to achieve.
When modeling the cage, keep it as close as possible to the geometry you want to deform.
When you are satisfied with the cage assignment, freeze the cage weights (that is, the Envelope_Weights operator under the CageWeightCls cluster).
