Thanks to all for the links above. I gather that what the overtone flutes do have in common with a natural trumpet is the reliance on overtones to get the notes of the scale, and perhaps also on bending those notes to vary their intonation. Of course they differ in how they get the column of air to vibrate. The slide whistle isn't along these lines i.e. a fixed-length column of air, rather it uses the slide to change the note by changing the length of vibrating air.

The overtone flute in any of its forms is a wonderfully simple instrument that obviously can be versatile under the right player.