Fractional diffusion on bounded domains is notorious for the lack of boundary regularity of solutions regardless of the smoothness of domain boundary. We explore this matter for the homogeneous Dirichlet problem for fractional-order quasi-linear operators with variable coefficients in Lipschitz domains and any dimensions; this includes fractional p-Laplacians and operators with finite horizon. We prove lift theorems in Besov norms which are consistent with the boundary behavior of solutions in smooth domains. The proof exploits the underlying variational structure and uses a new and flexible local translation operator. We further apply these regularity estimates to derive novel error estimates for finite element approximations of fractional p-Laplacians and present several simulations that reveal the boundary behavior of solutions.