We investigate complete minimal surfaces of finite total curvature such that their ends are foliated by closed lines of curvature. This condition on the ends is necessary if they have a piece inside some Euclidean ball that is free boundary. It turns out this is a rigid situation, and we are able to show, among further obstructions, that there are no such surfaces with one end. As a consequence, none of the orientable free boundary minimal surfaces with one boundary component and any genus discovered by M. Schulz, G. Franz and A. Carlotto can be obtained as a restriction of the Chen–Gackstatter surface or its generalizations to higher genus. Furthermore, in the non-orientable context, the Klein bottle of F. López and none of its generalizations to higher genus found by F. Martín with a single end can intersect a sufficiently large sphere in an orthogonal way.