I wonder really how connected in design any distros really are.
One upon a time, as far back as 2003, there was a distro called Slax (and still is) - originally used Slackware components, moved to Debian, and now does both I think
https://distrowatch.com/table.php?distribution=slax
https://www.slax.org/
and in 2010 a distro called Slax Remix became Porteus
http://www.porteus.org/
And from the Slax creator there was Linux Live scripts
https://www.linux-live.org/#changes
and from somewhere or other came Debian Live ... to be honest I've never been quite sure how that fitted in - but had its own initramfs, which used to be used in earliest Debian Dogs.
But I do know Tony (old forum member saintless, who left) originally used Debian Live initramfs to boot his Debian-based root filesystem (but he manually chopped down a Debian iso release back then).
Fred (fredx181) joined in early on and brought with him a modified version of Porteus initramfs to boot his preferred Openbox configurations (saintless was copying Puppy look and feel by adopting JWM/Rox) - used some systems out of Puppy Linux too - such as Puppy Volume monitor, though other ways of doing that such as gvfs with the likes of Thunar or Pcmanfm can be used instead. Rox also provided I think - but kind of secondary filemanager category once Fred took over I'd say with not so good mime support worked on.
Ability to load addons via modules was available from many distros (not just Puppy) so maybe more credit is due to Slax for the overall tricks?
End result, however, is that the DebianDogs have all the frugal install tricks (via Porteus boot) that Puppy itself provides via its own aufs-based initramfs. And for years now Debian Dogs have built their underlying main root filesystem via Debian-provided (or Ubuntu or Devuan) debootstrap scripts and bolted the modified Portues boot onto that. Fred has also since written tons of great utilities for improving the frugal install abilities (and adopted/modified some that were originally designed for Puppy Linux). So you end up with pretty much fully-compatible package management and root filesystem hierarchy as Debian itself (or Ubuntu or Devuan itself depending which debootstrap you started the build process with). And with all the usual Puppy frugal install type facilities - like Slax, and Porteus itself for that matter.
My own impression is that Thomas M of Slax deserves a lot of credit for such live distro designs that utilized aufs, but such designs were already ancient in the UNIX world with Sun microsystem using layered filesystem as far back as the 1980's...
https://en.wikipedia.org/wiki/UnionFS
Personally, I think it is fair to say that if you want true apt/dpkg capability from any distro on this forum the Debian Dogs are the best of breed, and you also get the full range of frugal install sfs-load and so on abilities too (though newer DebianDogs are using overlayfs so sfs-load is done by a different under-the-hood methodology. DebianDogs don't use PPM or pkg but they do have utilities that can convert dotpets into deb packages or sfs addons - and can use dpkg/apt to directly make sfs addons as an alternative to official distro package management - I think fredx181 created most of these nice utilities - a Puppy Linux user will find DebianDogs at least as easy to set up and use IMO - indeed I myself moved from Puppy Linux mainly to using the DDs back in 2013 and enjoyed the official dpkg/apt/Synaptics GUI package management provided, as well as the full proper multi-user capability. Helped me develop my own Linux abilities overall immensely. DebianDogs also shave the deb dependencies installed pretty well - and cut out a lot of docs and unwanted language packs I think (but all can be supplied in sfs addons, as can devx type sfs) - so pretty similar organisation to Puppy. However, so easy to install packages successfully in DD that it does tend to swell in size... easier to keep Puppy small in use I think - but maybe not if you install the same upstream repo packages! Reading people discussing Puppy Linux leads me to think that some believe that simple tar.gz based dotpets are the principal package type that Puppy is built from. That used to be the case, but a long time ago when it had its own repos - nowadays, for Debian/Ubuntu-based Pups, PPM is bringing down debs, converting them to format it can keep track off, and then installing them. Dotpets still hanging around out there though for some bits and pieces and user-designed utility apps. And now there is at last a move to adopting dpkg/apt itself in Puppy too.
Saintless no longer here to comment, and been Fred's work and development for many years so his view/comments will be interesting... Worth adding that KLV-Airedale user utilities owe a lot to Fred's additions that he ported across from his DebianDog work.