Kinder kommen nach dem Kindergarten in diese Schule.
This is a pretty straightforward use of a prepositional object giving a direction, as suggested (not strictly required) by the verb kommen. Nach dem Kindergarten in contary is an adverbial of time, because it's an addition not tightly connected to the verb kommen in particular.
Dies ist eine der ersten Einrichtungen, in die Kinder kommen.
This is way trickier. The word die doesn't belong to Kinder, instead, it's a backreference to Einrichtungen. Kinder has no article here, it's
"children as a kind". With article for Kinder:
Dies ist eine der ersten Einrichtungen, in die die Kinder kommen.
You may replace the first die with the relative pronoun welcher/welche/welches:
Dies ist eine der ersten Einrichtungen, in welche (die) Kinder kommen.
This construction preposition+relative pronoun (or article as a relative pronoun) to my knowledge is possible in English, too. But it seems to be bad style, English speakers seem to prefer a passive participle construction as this one:
This is one of the first instititutions attended by children.
While this is certainly possible in German, too,
Dies ist eine der ersten von Kindern besuchten Einrichtungen.
it sounds pretty "technical" in German. People don't talk like that and only write like that if they want to sound distant. The natural way in German is the relative clause lead by a preposition+article as a relative pronoun.