In this study, we first investigated a method for generating moving ILMs by deforming the Fourier spectrum of a stationary ILM in the 1D lattices with quartic nonlinearities in both on-site and inter-site potentials, which is called a Fermi--Pasta--Ulam--Tsingou--Nonlinear--Klein--Gordon (FPUT--NKG) mixed lattice. Secondly, the method for generating moving ILMs from a static ILM is extended for ILMs in 2D scalar square lattices. We will report the behaviors of the generated moving ILMs, which depend on the energy and the nonlinearity ratio in on-site and inter-site potentials.