Generally, Puppy has no automated update or upgrade feature.
Similar as in Windows you check yourself for newer versions of your software.
When you have a frugal installation you can upgrade some versions to their successors, such as Puppy 5.1 to 5.2. But e.g. 4.3 to 5.2 is a no go.
The best work around - in my opinion - is to make notes of your individual fine tuning of the basic Puppies and then repeat those in the next version. Keeping an archive of PETs and SFSs is also a good idea, as is the copying of modified files.
Firefox and Thunderbird can be updated through their build-in update function.
Sources for software information: