We investigate the off-equilibrium dynamics of spin systems with O(N) symmetry arising by the presence of slowly varying time-dependent external fields. We show the general theory and then focus on two different cases: a time-dependent magnetic field h(t,ts) ≈ t/ts, ts is a time scale, at the critical temperature and the temperature deviations T (t, ts )/Tc − 1 ≈ −t/ts in the absence of magnetic fields. We demonstrate the off-equilibrium scaling behaviours and formally compute the correlation functions in the limit of large N. We study the first deviations from the equilibrium in the correlation functions and prove that the matching occurs exponentially fast. We also consider analogous phenomena at the first-order transition which occurs in the ordered phase T < Tc along the line of zero magnetic field
