A pair of combined diffractive optical elements (DOEs) realizes a so-called moiré lens, with an optical power which can be tuned by a mutual rotation of the two DOEs around their central optical axis. Earlier demonstrated moiré lenses still suffered from chromatic aberrations. Here we experimentally investigate a multi-color version of such a lens, realized by a pair of multi-order DOEs. These DOEs have a deeper surface structure which modulates the phase of the transmitted light wave by several multiples of 2π. The corresponding multi-order moiré lenses all have the same focal length at a fixed set of harmonic wavelengths within the white light spectrum. The experiments demonstrate that multi-order moiré lenses have significantly reduced chromatic aberrations. We investigate the performance of the lens for narrow band and white light imaging applications.