I am looking for a way to customize Ash sessions with my own sets of
aliases and whatnots. What is the Ash equivalent of Bash's
A non-login shell will also read a file if specified in the environment variable ENV.
So if you set that somehow (Maybe in your ~/.profile, or some other 'overarching' environment control), then any future forked shells will run that script. Very handy for non-login cases.
It's hard to find documentation on this, but it is confirmed to work on busybox-w32 (running on Windows). In fact it's hard to find good documentation on the featureset of ash at all.
UPDATE: There are a range of ash variants in the wild. 'ENV' may not work with all of them. There is some info on variants here: https://www.in-ulm.de/~mascheck/various/ash/
There is a suggestion in there that some ash variants may use 'SHINIT' in place of ENV.