It is important to understand the difference between interface width and grid spacing in phase-field models. The interface width is a function of model parameters that enter the governing equations, and is not dependent on the grid spacing or other details of the discretization. For example, in the most basic Cahn-Hilliard model, the interface width is a function of the gradient energy coefficient $$$\kappa$$$ and the height of the free energy barrier between equilibrium phases. When you change the grid spacing and keep the system dimensions the same, you change the number of elements in the interfacial region, but not the width of the interface in the coordinate system you have chosen. So if you were to keep the system dimensions the same, simulate an interface between two phases with increasingly finer resolution, and plot the results on top of one another, you would see the same interface shape (width) represented with an increasing number of data points in the interfacial region, but the interface width in your coordinate system would not change.
If you keep the governing equations of the model the same, as you make the mesh finer, as long as the number of elements in the interface is high enough, you should get the same physical results. For the basic Cahn-Hilliard model I referred to earlier, a good rule of thumb would be that you would want at least 4-5 elements in the interface (defined as 0.1 < c < 0.9 if the equilibrium values of c are 0 and 1) if you are using linear Lagrange elements. You may be able to get away with fewer elements than that, and if you are looking only at qualitative differences after a few time steps, you may not notice any changes. But if you want to lower resolution below the 4-5 elements through the interface that I mentioned you probably should plot the system energy as a function of time to verify decreasing resolution is not affecting the system's evolution.