This work deals with the first Trefftz Discontinuous Galerkin (TDG) scheme for a model problem of transport with relaxation. The model problem is written as a PN or SN model, and we study in more details the P1 model in dimension 1 and 2. We show that TDG method provides natural well-balanced (WB) and asymptotic preserving (AP) discretization since exact solutions are used locally in the basis functions. High order convergence with respect to the mesh size in two dimensions is proved together with the asymptotic property for P1 model in dimension one. Numerical results in dimension 1 and 2 illustrate the theoretical properties.