We investigate interacting scenarios that belong to a wider class, since they include a dynamical dark energy component whose equation of state follows various one-parameter parametrizations. We confront them with the latest observational data from the cosmic microwave background, the joint light-curve sample from type Ia supernovae, baryon acoustic oscillations, Hubble parameter measurements from cosmic chronometers (CC), and a Gaussian prior on the Hubble parameter H0. In all examined scenarios we find a nonzero interaction; nevertheless, the noninteracting case is allowed within 2σ. Concerning the current value of the dark energy equation of state for all combinations of data sets, it always lies in the phantom regime at more than 2–3 standard deviations. Finally, for all interacting models, independently of the combination of data sets considered, the estimated values of the present Hubble parameter H0 are greater compared to the ΛCDM-based Planck estimate and close to the local measurements, thus alleviating the H0 tension.