We develops the structure-preserving finite volume weighted essentially non-oscillatory (WENO) hybrid schemes for the shallow water equations under the arbitrary Lagrangian-Eulerian (ALE) framework, dubbed as ALE-WENO schemes. The WENO hybrid reconstruction is adopted on moving meshes, which distinguishes the smooth, non-smooth, and transition stencils by a simple smoothness detector. To maintain the positivity preserving and the well-balanced properties of the ALE-WENO schemes, we adapt the positivity preserving limiter and the well-balanced approaches on static meshes to moving meshes. The rigorous theoretical analysis and numerical examples demonstrate the high order accuracy and positivity-preserving property of the schemes under the ALE framework. For the well-balanced schemes, it is successful in the unique exact equilibrium preservation and capturing small perturbations of the hydrostatic state well without numerical oscillations near the discontinuity. Moreover, our ALE-WENO hybrid schemes have an advantage over the simulations on static meshes due to the higher resolution interface tracking of the fluid motion.