Membranes...
The 'flukes and flippers' page explained how flippers in hexapods evolved form animals with mebranes on the sides of their bodies. Undulating waves travel backwards along the membrane, pushing against the water and thereby pushing the animal forwards. Some Terran fish use their dorsal or ventral fins in this way, and squids of course also use such undulating membranes. There is nothing new there.
The image here shows two stages in the transition from membrane-driven to flipper-driven protohexapods. The animal on the left is a rather unimpressive protohexapod, and the one on the right is more complex in shape. It is somewhere intermediate and is also shown on the 'flukes' page.
Let's move on cloakfish, the most iconic Furahan cloakfish swimmers.

How does swimming with membranes work?

The best known Furahan animals to rely on such a propulsion system are cloakfish, also featuring as one of the species in the Animals / Water pages. To the right another animation of cloakfish is seen. But no animal on Terra uses four such undulating membranes, which provides some intriguing possibilities for phase differences. In the animation, the cloakfish has its four cloaks in pahses that result in them moving towards one anther at certain points, explained in more detail below.

The screw gait
Yes, it is called that...
This model with four cloaks helps to explain phase differences. This particular movement has no phase differences, so the four cloaks carry out the same movements at each point along the dagger. The resulting movement looks a bit like turning in a screw, hence the name.

The 'walk'(?)
The four cloaks have intervals of one quarter phase between them. Try looking at the left edge only instead of trying to take the whole shape in. By analogy with phase differences between walking quadroupds (see that section), it's a walk!

The 'trot'
Those of you who worked through the 'walking with quadrupeds' page will know what a trot is. For circular structures the concept is less easy to visualise, but if you look hard, you may see that the four cloaks move in two pairs, and that the two pairs are in opposite phases, and the two cloaks forming one pair are opposite one another. It takes some thinking.
Anyway, you can see that the cloaks come close together, which helps enclose a volume of water and helps direct the propulsive force purely backwards. But not all adjacent cloaks move equally close together.

The 'pace'
Look longer and harder: again, the four cloaks move in two pairs, and again the two pairs are in opposite phases, but now the two cloaks forming one pair are next to one another, instead of opposite. Told you it takes some thinking about.
Now all adjacent cloaks touch their neighbours at certain points along their length, enclosing water and propelling it backwards at increased efficiency. Pacing appears to be the most effective cloakfish gait, at least for long-cloaked species ('shortsleaved cloakfish' are a different kettle of fish...