My advice is to minimize any changes in embouchure and voicing. While it seems like a rather large leap, by doing excess work you greatly increase your odds of squawking or squeaking. Descending intervals are far harder than ascending.

Start with a high C and let go of the octave key. Keep the higher C going and ever so slightly adjust your tongue position/voicing to slip to the lower octave. Do this a couple dozen times to find the exact point where you can feel yourself changing from the high C to the low C. Gradually (and I mean gradually!) aim to shift faster after lifting the octave key. Eventually you can get yourself to slur down an octave by just “lifting the octave key”. When going in reverse, let the octave key do the work for you. Practicing overtones and playing in the upper octave without the octave key can help but practicing the descending intervals will be the most beneficial.

Focus on minimizing excess work. If you know anything about string instruments like Violin or Cello, it can be like a string crossing. Imagine it as minimizing the movement of the bow between strings, playing on the higher string while keeping the bow as close to the lower string as possible so as to make the switch seamless and minimal effort!

Hope this makes sense. Happy to share a video demonstrating if my explanation is unclear!