If all their properties are randomized then I don't see how they can be said to "travel". That would be indistinguishable from electrons spontaneously appearing and unrelated electrons spontaneously disappearing.
This is one of the major reasons I said time travel without information transfer is not interesting. Even if it actually happens, it's not distinguishable from something else.
PS: Quantum entanglement is also less interesting than you might think for similar reasons.
In the case of entanglement though it is actually distinguishable from other, less "interesting" phenomena though, right? That's what the loophole-free Bell tests are about.
PS: Quantum entanglement is also less interesting than you might think for similar reasons.