Passing chord is more of a pop/jazz concept. Passing tone is the common practice period (CPP) concept that is usually discussed.
However, there is one particular chord that does come up in CPP harmony which gets labelled as "passing" and that is the passing 6/4 chord. The idea is 6/4, second inversion, chords are unstable and so are used only in limited way.
CPP also uses the label passing chord as a catch all for a vertical group of tones that you don't label with Roman numeral analysis or doesn't have any major significance in the overall harmony. Parallel, diatonic 6/3 chords is an example. "Non-functional" is another term used in the same way as "passing". You might see passages described as "non-functional seventh chords", and that would have the same meaning as "passing chord." The important thing here is, in regard to "strict rules", you won't find "rules" about this, because the concepts are applied to exceptional cases that don't follow the normal, functional practices. Think of it as labeling for moments of harmonic artistic license.
When comparing CPP and jazz harmony theory take note of the difference in analysis between CPP secondary harmony and jazz passing chords. Jazz harmony has a passing diminished chord concept which in CPP would be analyzed as a secondary dominant or leading tone chord.