  In the context of the future high energy - high luminosity electron and muon
colliders, all the relevant four-lepton processes with the lepton family
violation (LFV) are systematically classified. The most general LFV effective
lagrangians are found, and the helicity differential cross sections for the LFV
processes are calculated. The six- and eight-lepton Standard Model (SM)
backgrounds are discussed, and the LFV processes clean of the six-lepton
background are picked out. The possibility to suppress the six-lepton SM
background, when present, by the unnatural initial beam polarizations is
investigated. It is shown that the four-lepton LFV processes are amenable to
experimental study in the lepton-lepton collisions in the most favourable cases
up to the underlying scale of order 100 TeV. Studying these processes should
provide an essential part of the physics program for the next generation lepton
colliders to unravel the outstanding flavour/family problem.
