The principle of common-base operation is that we change voltage at the emitter, thus turning the transistor on and off.
I searched through my large collection of oscillator schematics which use one or two transistors. I had trouble finding one that illustrates common-base operation. In particular I was looking for a capacitor (or LC tank) in the emitter leg.
The closest one seems to be the emitter coupled multivibrator (below).
It has 2 NPN transistors, with a capacitor bridging the emitter legs.
The capacitor charges alternately in one direction, then the other. It changes the voltage at the emitter legs, alternately turning one transistor on, the other off.
The above was one of a bunch of multivibrators seen at:
https://www.4qdtec.com/mvibs.html
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Frequency can be tuned with the potentiometer, over a range of several percent.
If the pot resistance is too great, oscillations fade.