Does anyone know if this is a FBX thing?
Yes .FBX is a Skinned-mesh format, you need a Frame-mesh format like MD2.
FBX primary exports a mesh and then the bone transforms, it's your engine that then transforms the Skinnedmesh based on the bones imported from the .FBX.
There is a way to import the vertices animation using, morph targets (Blendshapes).
Note about morph targets: .FBX stores each morph target's final form as a mesh inside the file, then uses vector math for transitions.
Example: Vertex_A = (0,0,0) to Vertex_A = (5,0,0) so Morph target set to half will = Vertex_A (2.5,0,0) This is great for squash and stretch, however won't work for bends or rotations.
Also again your engine will do the animation meaning that if you just import it, the animation won't play by it'self. You will have to read up on how your engine uses Morphtargets.
A workaround is to convert each step of the animation to it's own morph target, import the now large .FBX file and convert each tatget to a frame in your engine specific Frame Animation Mesh.
MD2 is widely used for this as it has everything you need, materials should be made in your engine and not exported, MD2 doesn't hold to modern mesh standards and as such it is recommended you convert to your engine Framemesh.