Ich bin im Gebäude. = Ich bin darin.
Ich gehe ins Gebäude. = Ich gehe da rein.
Now that is cleared. My question is about other prepositions like an, neben, zwischen. How can one write for example daran with a moving object. For example:
Das Bild hängt an der Wand. = Das Bild hängt daran.
Ich hänge das Bild an die Wand. = Ich hänge das Bild ??daran??.
Does the preposition stay as "daran" or does it change?
No, there is only "daran". "darin" / "darein" is actually the special case in that it distinguishes the "location" and the "moving" sense. All other prenomial adverbs are used in both senses, as in
Ich hänge das Bild an die Wand. -> Ich hänge das Bild daran.
Ich stelle mich unter das Dach -> Ich stelle mich darunter.
Ich stehe unter dem Dach -> Ich stehe darunter.
Ich renne auf den Platz -> Ich renne darauf.
Ich renne auf dem Platz -> Ich renne darauf.
Similarly, woran, worunter etc. can both mean "an/unter welcher Sache" and "an/unter welche Sache".
A: The group of words earlier known as Präpositionaladverbia s.a.
darin, worin, hierin / daran, woran, hieran / damit, womit, hiermit
is these days usually called
Pronominaladverbia,
and rarely
Adverbialpronomina.
The earlier name "Präpositionaladverbia" is referring to the way they are built by combining an adverb (da, wo, hier) with a preposition (in, an, mit). (The former becomes a locative adverb, the latter a prepositional adverb; together, they form a pronomial adverb.)
The current names refer to the adverbs function, which is pronomial in the sense that their locative adverbs (da, wo, hier) refer to something mentioned earlier, the way many pronouns do.
Most grammaticians call them Pronominaladverben and classify them as adverbs, some call them Adverbialpronomina and classify them as Pronomina because of their proxy function, which is similar to pronouns' proxy function.
A: Indeed, all pronomial adverbs are invariant. This means that they are never subject to any declension morphologically, but they are, as seen above, syntactically dependent of what they refer to.
In your example
Ich bin im Gebäude. = Ich bin darin.
the pronomial adverb darin (or more precisely, its locative adverb da) refers to Gebäude, and its prepositional adverb in specifying the position.
Note however that when in bears the meaning of a motion ("into"), s.a. in your second example
Ich gehe ins Gebäude. = Ich gehe da rein.
then the usage of darin, darein, da rein is very rare; vastly more common here is hinein.
A: Your third and fourth examples arise around two different meanings of hängen, = to hang, one being a state, where the verb is intransitive:
Das Bild hängt an der Wand. = Das Bild hängt daran.
and the other one an action, where the verb is transitive:
Ich hänge das Bild an die Wand. = Ich hänge das Bild daran.
In both cases, daran is used correctly, and it (or any other Pronominaladverb) does not differ depending on the transitivity of the verb - or anything else for that matter, as they are, again, invariant.
Lastly, to quote your final question:
A: We have seen that they are not prepositions, but Pronominaladverben, = pronomial adverbs, and that No, they do not change, they are invariant.
