The shape morphing technique You are referrring to is known under multiple different names depending on the software and particular field of using it:
- Blend Shapes
- Morph Targets
- Shape Keys
- Corrective Shapes
In Blender they had been referred as "shape keys", although there is an ongoing discussion changing the name in 2.8 to "blend shapes".
In order for blend shapes to work you will always need two meshes not only with the same vertex count but also the same vertex order (the order in which the vertex data is written into the mesh data block). This way blender knows the difference between the two shapes and can interpolate accordingly. The general workflow is described in the blender manual:
Blender Manual Shape Keys
Just as a heads up: When importing mesh data it is possible the vertex order between meshes changes making it hard to blend between the original and edited version. This would be a different topic altogether and should normally work just fine using the "Join as Shapes" Tool as follows:
Select your imported object, SHIFT + RMB your target object, and in the Shape Keys section of your rigged object, hit the down arrow button for the drop down menu and select “join as shapes”. It will automatically create a “Basis” shape key, which is needed, and another shape key named after the imported object.