The generation of large-scale magnetic fields is generically accompanied by the more rapid growth of small-scale fields. The growing Lorentz force due to these fields backreacts on the turbulence to saturate the mean-field and small-scale dynamos. For the mean-field dynamo, in a quasilinear treatment of this saturation, it is generally thought that, while the alpha effect gets renormalized and suppressed by nonlinear effects, the turbulent diffusion is left unchanged. We show here that this is not true and the effect of the Lorentz forces is also to generate additional nonlinear hyperdiffusion of the mean field. A combination of such nonlinear hyperdiffusion with diffusion at small scales also arises in a similar treatment of small-scale dynamos, and is crucial to understand its saturation.