Let {u_n} be a sequence of maps from a compact Riemann surface Mwith smooth boundary to a general compact Riemannian manifold N withfree boundary on a smooth submanifold K ⊂ N satisfying

where τ(u_n) is the tension field of the map u_n. We show that the energyidentity and the no neck property hold during a blow-up process. Theassumptions are such that this result also applies to the harmonic mapheat flow with free boundary, to prove the energy identity at finite singulartime as well as at infinity time. Also, the no neck property holds at infinitytime.