Slow plasma flows perpendicular to the magnetotail neutral sheet and associated with flapping motions of the plasma sheet can occasionally exceed the sensitivity threshold of plasma instruments. This is the case in several events selected from 3 months of magnetotail observations by the three-dimensional plasma instrument on the AMPTE/IRM satellite. By comparing the ion flow in the ZGSM direction and the correlated changes of the XGSM-directed magnetic field, we estimate the current sheet flapping velocity to range from a few tens to a few hundreds of kilometers per second. Our measurements of the current sheet thickness (0.1-1RE) and current density (10-30 nA/m2) were significantly different from estimates of these parameters based on magnetospheric models, suggesting a stretched field configuration. The observed events took place during geomagnetically disturbed periods. One of those periods was a 2-hour-long convection bay interval. The largest flapping flows were observed during the few events which also exhibited high-speed Earthward flows, but the flapping velocity was smaller during the more common, slow convection intervals. This observation, as well as the unexpectedly large flapping velocities seen as close as 12RE (i.e., close to the hinge point), suggests that flapping motions originate near the current sheet. Plasma sheet flapping tends to occur in conjunction with ground Pi2 bursts, but the exact onset times and the dominant periods of the two phenomena are generally different. Thus, although a common energy source of both phenomena may be intermittent current disruption/reconnection, the two phenomena are likely to couple differently to that free-energy source. We discuss the vertical structure of the plasma sheet parameters during flapping plasma sheet episodes.