Going from meri to kari notes and vice versa requires a loose relaxed neck. The head needs to pivot quickly up and down on the neck, but with a minimum of tension. It helps if you think of the head as floppy, swinging on a hinge. For a good low meri just let it drop; for kari, swing it up. If you're trying to push the head tightly to the correct position you probably won't make it.
Here's an exercise utilizing some pairs of notes that students often play out of tune in the context of the traditional music, notes that require a very full swing from kari to meri and meri to kari. The first pair is ri no meri to i (ri with your thumb off), or in kan register hi no meri to go no hi. Using a pitch meter, make sure the lower note of the pair is all the way down to a B flat. Now swing your head way up as you remove the index finger and thumb of the left hand from the flute for a good high i (or go no hi). Many students play this out of tune. It should be the same pitch as ro, a western D. Go back and forth between these two notes, always making sure your ri no meri (or hi no meri) is low enough and your i (or go no hi) is high enough. Let your head feel heavy and relaxed as you swing between the two notes. Make sure that each is a discrete note -- in other words let your head move fast enough that there is no sliding from note to note. Visualizing a playground swing quickly rising and dropping helps with this.
The other pair of notes to try this with is tsu dai meri and re. Again, practice the back and forth in both otsu and kan registers, making sure that your tsu dai meri is all the way down to D, the same pitch as ro, and that your re is a good in tune G.
To get the feeling of a heavy head swinging on a relaxed neck, you can also try standing up, hanging over, and gently flopping your head as if nodding "yes."