Gratitude, Movement and Compassion
It seems to me that 'feeling moved' is just an emotional reaction. Meditation and practice are not about creating emotional reactions but about transcending them. The idea would be to establish who is moved, who is compassionate, who is emotional and so forth, as well as to engender compassion. 

I suspect gratitude is just what happens naturally but maybe this should also be practiced. Perhaps this would be the same as practicing 'being moved' but such a practice would also require establishing who is being moved and even what 'moved' really means.    

Experience without effort is not a practice but examining those experiences would be one. In short, I don't think there's anything in Buddhist practice one could call a free lunch. 

All 'in my opinion' of course.