chassee across

I've been looking at some nineteenth-century quadrilles and am curious about one of the more common moves I see: "all chassee across." In your opinion or experience, how exactly would this work? If four couples are moving at once, "across" is nearly impossible. Do you think they're doing a sashay around, like a promenade? Or alternating heads & sides? Something else? Regards,Deborah HylandSt. Louis