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.

11 pages, LaTeX, 1 PostScript figure. Some minor improvements and corrections are made