A great way to practice intervals for precise movement of fingers/hands and for flow of sound is by choosing an interval to work on and a rhythm. I wrote this exercise, Precision & Flow to demonstrate this. I used intervals that are tricky for us flutists. I also started with the right hand, then the left hand and both hands.
I encourage you to do these in front of a mirror and look for fingers to move from 3rd finger joint (near palm of hand) and that they do not bend or buckle. Here is link to free pdf https://www.creativeflutist.com/music
Also, I use this as an articulation exercise. I recommend that you slur it at first to make sure your breath and sound are flowing, then vary your articulations. As far as speed goes, start with ease and move into an edgy tempo that challenges you!
You will also benefit using your tuner and checking in on the long notes. I like to remember the quote attributed to A.. Marion who taught at The Paris Conservatory, "Never work for technique alone, always include sound"...
I have found this really helps my students and myself with their hand position, sound and technique in a fun and meaningful way.
Let me know how this works for you and your students.