I wouldn't use the term push per say however I do believe that we are capable of transmitting our thoughts, feelings, and ideas to other people. I would say it is a part of empathy, as an empath you connect with other people and on various levels you are able to sense and feel what they are feeling, thinking, and going through. With that said I believe it would taking your empathic abilities from a passive perspective and direct them towards a more assertive purpose.This is of course just my personal theory and what I believe; however, you can attribute this phenomenon to many other things such as telepathy, power of subtle suggestion, heck you can always just chalk it up to two people just clicking and being on the same wave length etc. I do know that influence is a powerful tool regardless of how it is done, some methods come with greater chances of success that others but that is likewise for most things.
For example you mentioned this happening between you and our grandfather. I personally believe and have experienced it myself that this type of mental connection whatever it may be called is more common amongst family members and people you have strong emotions for, such as best friends and people you have emotionally and even physically bonded with. Because of such is the reason I believe this type of mental connection is closer to assertive empathy above the rest.
Anyways I am fascinated by this subject and I hope you explore it further.