We present rigorous results on the exponential convergence to equilibrium for the Swendsen-Wang stochastic dynamics for the d-dimensional Ising ferromagnet with external magnetic field h in the thermodynamic limit. We consider various situations, mainly in the tow-temperature regime, in which boundary conditions are homogeneous and parallel or opposite to the external field. In the latter case we relate directly the tunneling from the metastable phase to the stable one with the exponential convergence to equilibrium.