Now that a spherical analogue of a non-degenerate plane Kempe network has been found, it might be expected that other, more general ones, will readily follow. We show here that such will not be the case, and that the incorporation of the Bennett linkage in the recently established spatial network is of fundamental significance to its mobility. We go on to highlight an error in a prior account of the Kempe linkages and to detail previously unrecorded relationships between the “type K” linkage and other chains featured in the kinematics literature. This latter part of the paper necessarily embodies an historical account of investigations engendered by the findings of Kempe and Hart.