We study dynamics of Dirac solitons in prototypical networks modeling them by the nonlinear Dirac equation on metric graphs. Stationary soliton solutions of the nonlinear Dirac equation on simple metric graphs are obtained. It is shown that these solutions provide reflectionless vertex transmission of the Dirac solitons under suitable conditions. The constraints for bond nonlinearity coefficients, conjectured to represent necessary conditions for allowing reflectionless transmission over a Y-junction are derived. The Y-junction considerations are also generalized to a tree and triangle network. The analytical results are confirmed by direct numerical simulations. Keywords: nonlinear Dirac equation, metric graphs, Lorentz transformation, Gross–Neveu model, Dirac solitons, reflectionless transport