The Lagrangian points are points of unstable balance (at least gravitationally; L4 and L5 are stable thanks to Coriolis force.), and that means an object not stabilized actively will fall out of them and start accelerating down the steepest gradient of whichever gravity well it toppled into.
But with the gravity wells being as irregular as they are...
and the object gaining momentum, never mind neither of the two bodies remaining at rest, the trajectory becomes less intuitive.
So, assuming a minimal nudge, just a smallest loss of stability, where would the object end up starting at different Lagrangian points? Which nudges out of L-points make the object crash into Earth or the Moon, which will let it escape the 2-body system, and will any set it on a mostly cyclic orbit? (possibly unstable due to tidal forces, but still one that will take some time to decay.)