Unstable interfaces are omnipresent in plasma processes in nature and technology at astrophysical and at molecular scales. This work investigates the interface dynamics with interfacial mass flux and focuses on the interplay of macroscopic and microscopic stabilization mechanisms, due to the inertial effect and the surface tension, respectively, with the destabilizing acceleration. We derive solutions for the interfacial dynamics conserving mass, momentum, and energy and find the critical values of the acceleration, density ratio, and surface tension separating the stable and unstable regimes. While the surface tension influences only the interface, its presence leads to the formation of vortical structures in the bulk. The vortical structures are energetic in nature, and the velocity field is shear free at the interface. We find that the conservative dynamics is unstable only when it is accelerated and when the acceleration value exceeds a threshold combining the contributions of macroscopic and microscopic mechanisms. In the unstable regime, the interface dynamics corresponds to the standing wave with the growing amplitude and has the growing interface velocity. For strong accelerations and weak surface tensions typical for high energy density plasmas, the unstable conservative dynamics is the fastest when compared to other instabilities; it has finite values of the initial perturbation wavelength at which the interface is stabilized and at which its growth is the fastest. We elaborate extensive theory benchmarks for experiments and simulations and outline its outcomes for application problems in nature and technology.