The purpose of this note is initially to present an elementary but surprising connectedness principle pertaining to the intersection of a fixed subvariety X of some ambient space Z with another subvariety Y which is ‘mobile’ (in the sense of being movable, rather than actually moving). It is via this mobility that monodromy enters the picture, permitting the crucial passage from ‘relative’ or total-space irreducibility to ‘absolute’ or fibrewise connectedness (and sometimes irreducibility). A general form of this principle is given in Theorem 2 below.