The Accelerator on a CHip International Program (ACHIP) funded by the Gordon and Betty Moore Foundation aims to demonstrate a prototype of a fully integrated accelerator on a microchip based on laser-driven dielectric structures until 2021. Such an accelerator on a chip needs all components known from classical accelerators. This includes an electron source, accelerating structures and transverse focusing arrangements. Since the period of the accelerating field is connected to the drive laser wavelength of typically a few microns, not only longitudinal but also transverse effects are strongly phase-dependent even for few femtosecond long bunches. If both the accelerating and focusing elements are DLA-based, this needs to be taken into account. In this work we study in detail the implications of a phase-dependent focusing lattice on the evolution of the transverse phase space of a transported bunch
