Exceptional points, which are spectral degeneracy points in the complex parameter space, are fundamental to non-Hermitian quantum systems. The dynamics of non-Hermitian systems in the presence of exceptional points differ significantly from those of Hermitian ones. Here, non-adiabatic transitions in non-Hermitian (Formula presented.) -symmetric systems are investigated, in which the exceptional points are driven through at finite speeds which are quadratic or cubic functions of time. Different transmission dynamics separated by exceptional points are identified, and analytical approximate formulas for the non-adiabatic transmission probabilities are derived. Possible experimental realizations with a (Formula presented.) -symmetric non-Hermitian 1D tight-binding optical waveguide lattice are discussed through non-Hermitian Bloch oscillations between different bands.