Mmm. First off, I'll skip over mobile enchanted items with ring target or duration effects on them. That's just too hard to think about at this time of night. :mrgreen:
The Dur: Ring description in the main book says the magus must actually trace out the ring, which to me means the ring has to rest on a solid surface - floating metal ring doesn't count. Floating carpet or floating disc probably would, in which case I could see it moving with the carpet/ Tenser's disc. I'd roughly say "the ring and the area inside the ring must rest on a solid, fairly flat surface at least as big as the ring. The surface may move, but if the ring moves relative to the surface the spell goes kaput". Now, if you rule the surface must be at least room or structure sized, you can disallow ring carpets but keep sailing ships or flying castles, respectively. This also means that if, say, the floor rots away in the middle of your ring, your ward would break.
Also important for floating rings, how far above or below the ring can spell effects go? It would feel very weird to me if you create a ward against faeries in your lab, and the merinita's faerie familiar on the next floor down suddenly can't enter a circular space in its lab. 
Ditto for faerie eagles flying a couple thousand feet up flying into your ward and bouncing off in midair! Probably a top limit of the smaller of 'nearest floor/ceiling' (for inside rooms) or some multiple of base individual size (for outdoor rings), and a bottom limit of 'whatever the ring is drawn on', so you'd have to levitate above the surface of the floating ring
So, ward on deck of ship, fine. Ward on flying carpet, moving table, or levitating gold platter mostly okay (honestly, a mobile "cage of (whatever)" seems like a really great idea for an enchanted item). A 'floating hoop of warding', no.
As far as "Circular Ward Against Puppies", what realm are they aligned with? Infernally cute?
Okay, simpler rule. Anything that would potentially make the ring move relative to any occupants, breaks the ring. Puppies on table, fine. Flip table over, ring breaks. Move the table sideways so fast that you pull it out from under the puppy, ring breaks. I thought that related to flooting hoops somehow, but I am too tired to figure out how.
Not really related, but one item I thought up for my ward magus was an "Instant circle". Enchant a metal ring to change size from "just right for your finger" to "standard size for warding circle". Portable, makes it easy to trace out a circle anywhere you go, and great for any hermetic geometers since its always a perfect circle.