how pitch moves
The tongue does the tuning
When you whistle with puckered lips, the opening stays roughly the same while the pitch changes. What moves is mostly your tongue. It reshapes the space inside your mouth, and that space acts like a small resonating chamber. A smaller space gives a higher note; a larger space gives a lower one.
You can feel this without whistling at all. Silently shape the vowel "oo" and then "ee". On "ee", the tongue rises and moves forward, shrinking the space. Whistle while making that same silent movement and the note will usually climb. Return towards "oo" and it falls.
This is why a steady single note is such a good foundation. Once the lips and breath are reliable, you can leave them alone and give your attention to one thing: the tongue.

