Recent advances in CubeSat micro-propulsion and low-energy cislunar trajectory design demonstrate that small satellites can indeed be self-propelled from Earth orbits to the Moon.
The retrieved literature contains multiple specific examples and mission architectures (such as EQUULEUS, LunaH-Map, and water-electrolysis propulsion studies) showing that CubeSats equipped with compact electric or chemical micro-propulsion can navigate from Earth orbits to the Moon or Earth-Moon Lagrange points using low-energy transfer manifolds.