
der Einzug
moving in
The process of moving into a new apartment or a new house.
Der Einzug in unsere neue Wohnung ist für nächsten Monat geplant.
Moving into our new apartment is planned for next month.
moving in / entry
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moving in
The process of moving into a new apartment or a new house.
Der Einzug in unsere neue Wohnung ist für nächsten Monat geplant.
Moving into our new apartment is planned for next month.

to find its way in
The introduction and establishment of something new, for example a technology, a fashion, or an idea, in a particular area.
Die künstliche Intelligenz hat in viele Bereiche unseres Lebens Einzug gehalten.
Artificial intelligence has found its way into many areas of our lives.

entry
The ceremonial entry into a place by an important person or a group, often as part of a ceremony or procession.
Der Einzug der siegreichen Mannschaft ins Stadion wurde von den Fans bejubelt.
The entry of the victorious team into the stadium was cheered by the fans.

collection
The collection or charging of money, especially of receivables, contributions, or taxes, often by direct debit.
Die Bank ist für den Einzug der monatlichen Kreditraten verantwortlich.
The bank is responsible for collecting the monthly loan installments.
'Der Einzug' (moving in / entry) is the noun related to the separable verb 'einziehen' (to move in / to enter). The prefix 'ein-' here means movement into something, which explains the noun's different meanings.
Nächste Woche ziehen wir in die neue Wohnung ein.
Next week we are moving into the new apartment.
Since the word describes movement or entry, it is very often used with the preposition 'in' and the accusative: der Einzug in die Wohnung (moving into the apartment), der Einzug ins Finale (entry into the final).
Der Einzug in die Endrunde war eine große Überraschung.
Making it into the final round was a big surprise.
When it is about moving into an apartment, words like 'Wohnung' (apartment), 'Haus' (house), 'Umzug' (move), 'Möbel' (furniture), or 'Kisten' (boxes) often appear in the context.
Wir müssen für den Einzug noch viele Kisten packen.
We still have to pack a lot of boxes for the move-in.
The abstract meaning 'introduction of something new' is almost always used in the fixed expression 'Einzug halten' (to find its way in / to become established). The subject is the new thing, e.g. a technology or idea.
Das Internet hat schnell in fast allen Haushalten Einzug gehalten.
The internet quickly found its way into almost all households.
You can recognize the meaning 'ceremonial entry' from words like 'feierlich' (ceremonial), 'König' (king), 'Mannschaft' (team), 'Stadion' (stadium), or 'Parade' (parade). It describes a ceremonial act.
Der feierliche Einzug des Brautpaares in die Kirche begann.
The ceremonial entry of the bride and groom into the church began.
When it is about collecting money, terms like 'Bank' (bank), 'Konto' (account), 'Lastschrift' (direct debit), 'Gebühren' (fees), or 'Steuern' (taxes) often appear in the sentence.
Der monatliche Einzug der Miete erfolgt immer am Monatsanfang.
The monthly collection of the rent always takes place at the beginning of the month.
move / parade
Both words refer to moving to a new home. “Der Umzug” means the entire move, often including packing and transportation; “der Einzug” emphasizes moving into and arriving at the new home. Use “Umzug” for the overall process and “Einzug” for the beginning at the new location.

Unser Einzug in die neue Wohnung ist am Samstag.
We are moving into the new apartment on Saturday.
Unser Umzug in die neue Wohnung dauert das ganze Wochenende.
Our move to the new apartment will take the whole weekend.