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By a ring board, do you mean what is used for Jumping Chess and Rococo?

The boards lie about as flat as the surface you lie them on. Parts of the
board become elevated and tilted only when you fold the board into other
dimensions by folding part of the board underneath the rest of the board.
As a test, I just folded my 12x12 board into an 8x8 board on my bed, whose
surface is a memory foam mattress pad, and set up some small hollow plastic
pieces. Several of them fell down. I then removed that board and placed my
8x8 Cavalier Chess board in the same place, and set up the same pieces. I
shook my bed, which has springs underneath, and the pieces remained
standing. As long as you lie the board on a flat surface, such as a table,
and don't fold the board, it should be as flat and stable as you need it
to be. If folding the board makes it too unstable for your pieces, then
you can make more boards of smaller sizes, which you don't have to fold.