  Here we perform the Kaluza-Klein dimensional reduction from $D+1$ to $D$
dimensions of massless Lagrangians described by a symmetric rank-2 tensor and
invariant under transverse differmorphisms (TDiff). They include the linearized
Einstein-Hilbert theory, linearized unimodular gravity and scalar tensor
models. We obtain simple expressions in terms of gauge invariant field
combinations and show that unitarity is preserved in all cases. After fixing a
gauge, the reduced model becomes a massive scalar tensor theory. We show that
the diffeomorphism (Diff) symmetry, instead of TDiff, is a general feature of
the massless sector of consistent massive scalar tensor models. We discuss some
subtleties when eliminating St\"uckelberg fields directly at action level as
gauge conditions. We also show that the reduced models all have a smooth
massless limit.
  A non local connection between the massless sector of the scalar tensor
theory and the pure tensor TDiff model leads to a parametrization of the non
conserved source which naturally separates spin-0 and spin-2 contributions in
the pure tensor theory.
  The case of curved backgrounds is also investigated. If we truncate the non
minimal couplings to linear terms in the curvature, vector and scalar
constraints require Einstein spaces as in the Diff and WTDiff (Weyl plus Diff)
cases. We prove that our linearized massive scalar tensor models admit those
curved background extensions.
