Diffusion on 2D site percolation clusters at p=0.7, 0.8, and 0.9 above pc on the square lattice in the presence of two crossed bias fields, a local bias B and a global bias E, has been investigated. The global bias E is applied in a fixed global direction whereas the local bias B imposes a rotational constraint on the motion of the diffusing particle. The rms displacement Rt~ tk in the presence of both biases is studied. Depending on the strength of E and B, the behavior of the random walker changes from diffusion to drift to no-drift or trapping. There is always diffusion for finite B with no global bias. A crossover from drift to no-drift at a critical global bias Ec is observed in the presence of local bias B for all disordered lattices. At the crossover, value of the rms exponent changes from k=1 to k<1, the drift velocity vt changes from constant in time t to decreasing power law nature, and the ``relaxation'' time τ has a maximum rate of change with respect to the global bias E. The value of critical bias Ec depends on the disorder p as well as on the strength of local bias B. Phase diagrams for diffusion, drift, and no-drift are obtained as a function of bias fields E and B for these systems.