apng / mng never gained much support. ffmpeg can decode animated png, but can only create animated gif and webp. (No other ffmpeg output formats support a loop flag in the container, as far as I can tell, not even nut, mkv, or ogg.)
webp is the current animation-supporting image format that's trying to gain traction. With google behind it, there's a good chance. It can do lossy and lossless, and even lossy-image with lossless-transparency. It's based on the intra-frames of the VPx video codec, IIRC.
ffmpeg can create webp animations, but even ffplay can't play them back. vwebp (in the webp package) can.
ffmpeg -framerate 15 -i b93-'%d.png' -loop 128 containerloop.webp
The other option for putting repeating loops of images on the web is an HTML5 video tag with the loop attribute. Great for higher resolution, and live action (rather than computer graphics with a lot of areas all the same color). Don't use it all over the place instead of animated gifs, though. Web browsers aren't optimized for lots of small videos on pages.