Progress in information processing relies on spintronics, where magnetic states serve as efficient carriers for data storage and transfer. In this work, we theoretically study magnon propagation in a bilayer composed of a ferro-magnet and an antiferromagnet. For this purpose, we probe the spin Seebeck effect by introducing a spatially varying temperature profile. This generates a local magnon excitation and a continuous magnon flux from hot to cold regions which we quantify through the resulting non-equilibrium magnon accumulation. Based on the chirality of these modes, we identify specific constraints for magnon modes traveling either from the ferromagnet into the antiferromagnet or vice versa. A key finding is the observation of a thermally triggered spin current in the antiferromagnet, a phenomenon typically absent in bulk antiferromagnets that obey time-reversal symmetry. These results provide important insights into the design of heterostructures for magnonic chirality-selective spin transport.