Triggered by this question.
What's the difference in meaning between:
Stantes erant pedes nostri, in atriis tuis, Jerusalem.
Stabant pedes nostri, in atriis tuis, Jerusalem.
and does this apply broadly throughout the language?
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There is no difference in meaning, but it's rare to find a present participle used, as here, in a periphrastic conjugation : such usage is quite legitimate, but isn't normally recommended for prose composition. There is a perfectly good imperfect tense in Latin which supplies the need, as in your second version. Your example is from Ps. 121 in the Vulgate, where idiosyncratic grammar is not unusual.
To answer your general point : periphrastic conjugations in Latin are almost entirely restricted to two types.
The active future participle with sum as auxiliary verb: using from your example the verb sto, sample forms are staturus sum, I am about to stand; staturi erant, they were about to stand. This is the active periphrastic conjugation.
The gerundive is used in a similar way to form a passive periphrastic conjugation. Examples are monendus est, he ought to be advised, and monendi eratis, you ought to have been advised.