"Mindfulness" in Buddhism is generally a reference to "Four Stations of Mindfulness" (both in Theravada and Mahayana). It means you should practice to be mindful of Body, Mind, Feelings and Dharmas all time. Our minds are extremely programmable, so if you practice (put effort, use concentration) to be "mindful" for quite some time, then you will automatically get accustomed to be more mindful in everyday life effortlessly.
In our normal everyday life, we get drawn to random everyday thoughts and get carried away by them and this is our mundane/natural state because we have been doing this since a very long time. But if we try to be mindful, our normal state of mind would shift to the state of being mindful all the time.
I am not sure, though, that if you are mindful all the time does it also include "grasping" at dharmas all the time? Or in other words, is it sufficient if you do not grasp at Dharmas and therefore naturally come to a mindful state?
In my experience, animals show a very great deal of mindfulness. Observe cats, leopards, Tigers etc, and they never can be caught off guard!
No grasping to mental constructs, people can get fear by thinking their practice will starve itself to death. But resting in simplicity, free of taking things as real, well