We study the two-color nonreciprocal effects of transmission and reflection in cold atoms driven into the tripod configuration and confined in a moving optical lattice. Our numerical results show that a very high contrast of the forward-backward transmission up to around 92% (reflection up to around 85%) is observable near the sharp edges of two tunable photonic band gaps at lattice speeds of several meters per second. Such two-color optical nonreciprocity is attained in fact by breaking the time-reversal symmetry with asymmetric Doppler shifts and can be dynamically manipulated by varying the driving and coupling field detunings, the probe pulse length, the atomic lattice velocity, etc.