This work presents and analyses a new arbitrary Lagrangian-Eulerian (ALE) surface formulation for area-incompressible Navier-Stokes flow on self-evolving manifolds [1, 2]. The formulation is physically frame-invariant, applicable to large deformations, and relevant to fluidic surfaces such as soap films, capillary menisci and lipid membranes. Its unknowns are the fluid velocity, fluid pressure and surface mesh motion. The resulting three-field system is monolithically coupled, and fully linearized within the Newton-Rhapson solution method. It is implemented in a nonlinear surface finite element framework using the pressure stabilization scheme of [3] and the mesh stabilization scheme of [4], which are all adapted here to the new ALE frame. The unknown ALE mesh motion is determined by membrane elasticity such that the in-plane mesh motion is stabilized without affecting the physical behavior of the system. The new formulation is demonstrated on several challenging examples including shear flow on self-evolving surfaces, vesicle budding, and inflating soap bubbles with partial inflow on evolving boundaries. Optimal convergence rates are obtained in all cases. Particularly advantageous are C1-continuous surface discretizations, for example based on NURBS. [1] Sauer, R.A., A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds – General theory and analytical solutions, J. Fluid Mech., 1016:A34, 2025. [2] Sauer, R.A., A curvilinear surface ALE formulation for self-evolving Navier-Stokes manifolds – Stabilized finite element formulation, Comput. Methods Appl. Mech. Engrg., 447:118331, 2025. [3] Dohrmann, C.R., Bochev, P.B., A stabilized finite element method for the Stokes problem based on polynomial pressure projections, Int. J. Numer. Meth. Fluids, 46:183-201, 2004. [4] Sauer, R.A., Stabilized finite element formulations for liquid membranes and their application to droplet contact, Int. J. Numer. Meth. Fluids, 75(7):519-545, 2014.