Despite considerable research on document spanners, little is known about the expressive power of generalized core spanners. In this paper, we use Ehrenfeucht-Fraïssé games to obtain general inexpressibility lemmas for the logic FC (a finite-model variant of the theory of concatenation). Applying these lemmas give inexpressibility results for FC that we lift to generalized core spanners. In particular, we give several relations that cannot be selected by generalized core spanners, thus demonstrating the effectiveness of the inexpressibility lemmas. As an immediate consequence, we also gain new insights into the expressive power of core spanners.