I'm not a teacher, I learned to play legato when I was a kid, and I'm an amateur with perhaps not the greatest technique so take the below with a grain of salt.
I played slow legato and watched my fingers. One finger releases at the same time the other finger presses. The notes sound connected but not blurred. I took a video but I'm not going to share it here because I think you'd be better off watching a video on playing legato from a piano teacher on youtube.
When I play I move my elbow and wrist around. But I don't think I'm doing it for legato. My elbow and wrist also move when I play detached, albeit somewhat differently. The reason I'm moving my elbow and wrist is to get my fingers in the right place so I can play evenly. Having the elbow and wrist involved is critical for playing evenly at a faster tempo.