Specifically, loss of time, distance and size. Also advanced forms of empathy and being in two locations at once. I realise this is quite vague but I hesitate at giving details. I'm more concerned with how these are addressed in Buddhism.
Not really sure what type of answer you are looking for (sutra references about supernormal powers??), but I'll just paraphrase what I heard recently from a Theravada monk that seemed like extremely good advice which reportedly came from Ajahn Chah...
The advice was to simply be mindful and beware growing attached to any weird experiences you have while meditating. With the deep concentrations found in meditation you can have all kinds of seemingly supernatural experiences. So what. They are no more real or inherently existing (nor less!) than any thing experienced in waking life. If you have weird or seemingly supernatural experiences in waking life... again, so what. They are no more real (nor less!) than what you might experience in a dream. All things lack inherent existence and are not worthy of grasping. Grasp at them and you will incur suffering for yourself and others. So don't do it! :)
Now, the one thing you said that I think might deserve extra attention is your remark about "advanced forms of empathy." If by "empathy" you mean compassion (the wish for others to be free from suffering), then this is an unalloyed virtue that should be cultivated to perfection. To do so, I'd suggest Shantideva's Guide and loving kindness or bodhicitta meditation.
Hope this helps!