We study boundaries in N = (2, 2) supersymmetric theories in two dimensions. These models are of great importance to string compactifications from 10 to 4 dimensions, because they can be used to describe the dynamics of the 6 dimensional internal space of the target spacetime. We discuss D-branes in the internal dimensions, focusing on D-branes preserving B-type supersymmetry. We are particularly interested in the behaviour of D-branes under transport through the moduli space of a class of Calabi-Yaus with one Kaehler parameter. To study the behaviour we use the hemisphere partition function of the gauged linear sigma model to compute D-brane charges and monodromy matrices.