By means of free fermionic techniques combined with multiple precision arithmetic we study the time evolution of the average magnetization, m(t), of the random transverse-field Ising chain after global quenches. We observe different relaxation behaviors for quenches starting from different initial states to the critical point. Starting from a fully ordered initial state,
the relaxation is logarithmically slow and in a finite sample of length L the average magnetizatio saturates at a size-dependent plateau. Starting from a fully disordered initial state, the magnetization stays at zero for a period of time until t=t_d, which is exponentially large in l, l being the square root of L, and then starts to increase until it saturates to an L-dependent asymptotic value. For both quenching protocols, finite-size scaling is satisfied in terms of the scaled variable ln t/l. Furthermore, the distribution of long-time limiting values of the magnetization shows that the typical and the average values scale differently and the average is governed by rare events. The non-equilibrium dynamical behavior of the magnetization is explained through semi-classical theory.