Gravitomagnetism is mainly attributed to rotating mass (e.g. the Lense-Thirring effect). On the other hand it is said that gravitomagnetism is referring to kinetic energy and to movements of mass, that it is proportional to the velocity of the exposed particle.
So, what about radially free-falling particles (at velocity $v$) within the field of a non-rotating black hole? Are they exposed to some gravitomagnetic acceleration, from the point of view of an external observer?
For a non-rotating black hole the answer is no, in the rest frame of the hole, because the hole is not moving so does not produce a gravitomagnetic field.
For a rotating black hole (Kerr metric) there are gravitomagnetic effects for infalling particles.
For a non-rotating black hole in a frame where the hole is moving there will be gravitomagnetic effects. The analogy with the electromagnetic case of a moving point charge gives a gives a good first intuition; you just have the worry about the sign and magnitude.
