This study presented a three-dimensional (3D) finite element method (FEM) for the numerical analysis of fluid flow in domains containing moving interfaces. This method falls into the general category of Arbitrary Lagrangian Eulerian (ALE) method; based on a fixed mesh that is locally adapted at the moving interfaces and reverts to its original shape once the moving interfaces go over the elements. The 3D domain occupied by the fluid at any time in the simulation is used as the reference domain and is discretized using a mesh of hexahedral tri-linear isoparametric finite elements. The moving interfaces are defined by sets of marker points so that the global mesh is independent of interface movement and eliminates the possibility of mesh entanglement. The mesh never becomes unsuitable due to its continuous deformation, thus eliminating the need for repeated re-meshing and interpolation. A validation is presented via a problem with an analytical solution for the 3D flow between two planes separating at a prescribed speed that shows second order accuracy. The model’s capabilities are illustrated through application to laminar incompressible flows in different geometrical settings that show the flexibility of the technique.