[en] Here, we present a general concept to accelerate non-relativistic charged particles. Our concept employs an adiabatically-tapered dielectric-lined waveguide which supports accelerating phase velocities for synchronous acceleration. We propose an ansatz for the transient field equations, show it satisfies Maxwell's equations under an adiabatic approximation and find excellent agreement with a finite-difference time-domain computer simulation. The fields were implemented into the particle-tracking program (\sc astra) and we present beam dynamics results for an accelerating field with a 1-mm-wavelength and peak electric field of 100~MV/m. The numerical simulations indicate that a ∼200-keV electron beam can be accelerated to an energy of ∼10~MeV over ∼10~cm. The novel scheme is also found to form electron beams with parameters of interest to a wide range of applications including, e.g., future advanced accelerators, and ultra-fast electron diffraction.