Transport phenomena arising from the up-down asymmetry of ripple and helical perturbations of the magnetic field in tokamaks and stellarators are investigated. Accounting for this asymmetry is important for devices with vertically displaced magnetic axes or with a single null divertor. The asymmetry manifests itself in different magnitudes of the ripple or helical perturbations in the up and down parts of the magnetic surface. Owing to this difference, the toroidally trapped particles acquire a net radial drift. The collisionless dynamics of the particles in such configurations are studied, as well as the solutions of the kinetic equation. It is shown that, in spite of the net radial drift, the up-down asymmetry does not lead to enhanced diffusive transport, compared with the conventional ripple transport, although the convective loss at low collisionality might be large