  We explore general scalar-tensor models in the presence of a kinetic mixing
between matter and the scalar field, which we call Kinetic Matter Mixing. In
the frame where gravity is de-mixed from the scalar this is due to disformal
couplings of matter species to the gravitational sector, with disformal
coefficients that depend on the gradient of the scalar field. In the frame
where matter is minimally coupled, it originates from the so-called beyond
Horndeski quadratic Lagrangian. We extend the Effective Theory of Interacting
Dark Energy by allowing disformal coupling coefficients to depend on the
gradient of the scalar field as well. In this very general approach, we derive
the conditions to avoid ghost and gradient instabilities and we define Kinetic
Matter Mixing independently of the frame metric used to described the action.
We study its phenomenological consequences for a $\Lambda$CDM background
evolution, first analytically on small scales. Then, we compute the matter
power spectrum and the angular spectra of the CMB anisotropies and the CMB
lensing potential, on all scales. We employ the public version of COOP, a
numerical Einstein-Boltzmann solver that implements very general scalar-tensor
modifications of gravity. Rather uniquely, Kinetic Matter Mixing weakens
gravity on short scales, predicting a lower $\sigma_8$ with respect to the
$\Lambda$CDM case. We propose this as a possible solution to the tension
between the CMB best-fit model and low-redshift observables.
