We present a scattering model for nuclei with similar masses. In this three-body model, the projectile has a core+valence structure, whereas the target is identical to the core nucleus. The three-body wave functions must be symmetrized for the exchange of the cores. This property gives rise to non-local potentials, which are computed without approximation. The present model is an extension of the Continuum Discretized Coupled Channel (CDCC) formalism, with an additional treatment of core exchange. We solve the coupled-channel system, including non-local terms, by the $R$-matrix method using Lagrange functions. This model is applied to the $^{13}{\rm C}+^{12}$C, $^{13}{\rm N}+^{12}$C and $^{16}{\rm O}+^{12}$C systems. Experimental scattering cross sections are fairly well reproduced without any parameter fitting. The backward-angle enhancement of the elastic cross sections is due to the non-local potential. We discuss in more detail the various non-local contributions and present effective local potentials.

13 pages, 11 figures, submitted to Physical Review C