Its not guaranteed to work and only a thought, but try having a cold bowl / bucket of water next to your press, as soon as your done put the cup into the bucket of water, this should close up the "pores" and hopefully prevent any ink travelling, if this doesnt work then maybe try a different mug supplier? Ive had issues with the cheaper mugs, decided to pay an extra 30p a mug for the quality, Ive just added 15p onto my prices and ill cover the other 15p from my profits, if its a case of losing customers then its alwas worth it.
Hope the bucket of water tip helps you out, a brief reasoning why this MAY help is as followed:
The poly coating on the mug is somewhat like skin, when it heats up, it opens up (like a humans pores would) and naturally when it cools down the close. If the pores remain open there is a chance of ink spreading upwards so with a bucket of water you can close them very quickly.
Another reason this may occur (again just a thought) may be due to too much or too little pressure.
Anyway good luck try these tips and my fingers are crossed for you.