This means that either a particle has to completely 'vanish' from the physical world during tunnelling or that μ and ε don't apply during this time, or that the

This means that either a particle has to completely 'vanish' from the physical world during tunnelling or that μ and ε don't apply during this time, or that they change value. The standard quantum mechanical interpretation is that the particle does not have a single position during tunneling. Instead, its position is a distribution best thought of as a wave, or as a function of an underlying wave. You can interpret this as "many worlds" or you can interpret it as "measurement matters", you get the same results either way. The speed of the particle is also not a single value, but rather a different function of the same underlying wave.