Default FBX won't export the changes in the geometry - it will give you the position, rotation and scale of the object over time but not changes to vertex positions within the object.
You can turn on the 'Geometry Cache' option in the fbx export dialog:
Vertex Animation stored in FBX file without using Point Cache?
However not all applications that use FBX support geometry caches - you'll have to see what your target software supports. Other options might be to export a series of .OBJ files (1 per frame) and use the vertex data from those, or to use an animated geometry specific format like Alembic