  Various authors have investigated the problem of light deflection by radially
moving gravitational lenses - with inconsistent results. In this paper we
generalize the calculations for arbitrary lens velocities and show that, to
first order in the lens velocity v, the deflection angles scales with 1-v. We
discuss the seeming inconsistency of relativistic light deflection with the
classical picture of moving test particles by generalizing the lens effect to
test particles of arbitrary velocity, including light as a limiting case. We
show that the effect of radial motion of the lens is very different for slowly
moving test particles and light and that a critical test particle velocity
exists for which the motion of the lens has no effect on the deflection angle
to first order. An interesting and not immediately intuitive result is obtained
in the limit of a highly relativistic motion of the lens towards the observer,
where the deflection angle of light reduces to zero. This phenomenon is
elucidated in terms of moving refractive media. Furthermore, we discuss the
dragging of inertial frames in the field of a moving lens and the corresponding
Lense-Thirring precession, in order to shed more light on the geometrical
effects in the surroundings of a moving mass. In a second part we discuss the
effect of transversal motion on the observed redshift of lensed sources. We
demonstrate how a simple kinematic calculation explains the effects for
arbitrary velocities of the lens and test particles. Additionally we include
the transversal motion of the source and observer to show that all three
velocities can be combined into an effective relative transversal velocity
similar to the approach used in microlensing studies.
