  The aim of this paper is to present an abstract and general approach to jump
inequalities in harmonic analysis. Our principal conclusion is the refinement
of $r$-variational estimates, previously known for $r>2$, to end-point results
for the jump quasi-seminorm corresponding to $r=2$. This is applied to the
dimension-free results recently obtained by the first two authors in
collaboration with Bourgain and Wr\'obel (arXiv:1708.04639 and
arXiv:1804.07679), and also to operators of Radon type treated by Jones,
Seeger, and Wright.
