We use new and archival Chandra and ROSAT data to study the time variability of the X-ray emission from the pulsar wind nebula (PWN) powered by PSR B1509-58 on timescales of 1 week to 12 yr. There is variability in the size, number, and brightness of compact knots appearing within 20'' of the pulsar, with at least one knot showing a possible outflow velocity of ~0.6c (assuming a distance to the source of 5.2 kpc). The transient nature of these knots may indicate that they are produced by turbulence in the flows surrounding the pulsar. A previously identified prominent jet extending 12 pc to the southeast of the pulsar increased in brightness by 30% over 9 yr; apparent outflow of material along this jet is observed with a velocity of ~0.5c. However, outflow alone cannot account for the changes in the jet on such short timescales. Magnetohydrodynamic sausage or kink instabilities are feasible explanations for the jet variability, with timescales of ~1.3-2 yr. An arc structure located 30''-45'' north of the pulsar shows transverse structural variations and appears to have moved inward with a velocity of ~0.03c over 3 yr. The overall structure and brightness of the diffuse PWN exterior to this arc and excluding the jet has remained the same over the 12 yr span. The photon indices of the diffuse PWN and possibly the jet steepen with increasing radius, likely indicating synchrotron cooling at X-ray energies.