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.

'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.
cover / reference
'der Bezug' and 'der Einzug' can both appear in business or organizational contexts, especially when something is delivered or paid for regularly. 'der Bezug' means the regular receipt of a product or service, for example a newspaper subscription. 'der Einzug', on the other hand, means the collection of money, often automatically from an account. The words can be confused because they sound similar and both have to do with regular processes. The direction is important: with 'Bezug', you receive something; with 'Einzug', money is collected.

Der Einzug des Mitgliedsbeitrags erfolgt immer am Anfang des Monats.
The membership fee is always collected at the beginning of the month.
Der Bezug der Zeitschrift endet automatisch nach einem Jahr.
The subscription to the magazine ends automatically after one year.