Most Paul Jones-style mixers are similar to the one found in the manual Dancing Made Easy (New York, c1919-1922): the dancers form a circle of couples, or possibly two concentric circles, each operating separately, if the numbers are too great. They perform a grand right and left, then at the leader's signal dance a one-step with the next person in the chain. The 1903 Round Two-Step (described here) is quite similar, except that the dance of choice is a two-step.
But in the mid-1920s English manual Foulsham's Modern Dancing, by Maxwell Stewart, a more elaborate version is described.
To initiate the dance, the band plays a march and the dancers begin the grand right and left as usual. The signal to change to dancing in pairs is given by a change of music: the band switches to a foxtrot tune. Each dancer takes the next partner in the chain, and they foxtrot, with Stewart noting that the steps used are "as a rule" limited to the walk and the "two-step Chassé to right," though the latter is not actually given in his list of foxtrot steps earlier in the book and is more reminiscent of 1910s foxtrot style (see my previous post on this here) than the steps he does give, which look more towards later style. Perhaps that is intentional, on the assumption that everyone knows the "old-fashioned" steps but is less likely to know the newer ones.
At some point, the music changes back to a march, and they resume the grand right and left. Once again the music changes, this time to a one-step, and the dancers take next partners for that dance. The music shifts back to a march a final time for a third grand right and left until finally the band begins to play a waltz and the dancers each take the next partner and end the Paul Jones by waltzing.
Stewart positively burbles with enthusiasm about the medley aspect:
It is a very jolly affair for those who can execute the steps of the three present-day dances. It is a pot-pourri of all that is best at the moment, and our only wonder is that more medley dances are not evolved.
His initial description of the dance is dryer and more humorous, possibly unintentionally:
It is the nearest thing we have now to the old square dances, with just this difference: that it is round.
Indeed it is.