  We place observational constraints on two models within a class of scenarios
featuring an elastic interaction between dark energy and dark matter that only
produces momentum exchange up to first order in cosmological perturbations. The
first one corresponds to a perfect-fluid model of the dark components with an
explicit interacting Lagrangian, where dark energy acts as a dark radiation at
early times and behaves as a cosmological constant at late times. The second
one is a dynamical dark energy model with a dark radiation component, where the
momentum exchange covariantly modifies the conservation equations in the dark
sector. Using Cosmic Microwave Background (CMB), Baryon Acoustic Oscillations
(BAO), and Supernovae type Ia (SnIa) data, we show that the Hubble tension can
be alleviated due to the additional radiation, while the $\sigma_8$ tension
present in the $\Lambda$-Cold-Dark-Matter model can be eased by the weaker
galaxy clustering that occurs in these interacting models. Furthermore, we show
that, while CMB+BAO+SnIa data put only upper bounds on the coupling strength,
adding low-redshift data in the form of a constraint on the parameter $S_8$
strongly favours nonvanishing values of the interaction parameters. Our
findings are in line with other results in the literature that could signal a
universal trend of the momentum exchange among the dark sector.
