Falls der Einzug sich verschiebt, bleiben wir für ein paar Tage bei meiner Mutter.

Falls means if.

In many sentences, falls and wenn can both be translated as if, but there is a nuance:

• falls = more clearly conditional, often a bit more formal or written
• wenn = more general and very common in everyday German; it can also mean when in other contexts

So:

• Falls der Einzug sich verschiebt ... = If the move-in gets delayed ...
• Wenn der Einzug sich verschiebt ... would also be natural

Using falls emphasizes that this is a possible situation, not a certain one.

Because falls introduces a subordinate clause. In German subordinate clauses, the conjugated verb goes to the end.

So:

• Falls der Einzug sich verschiebt
  • falls = subordinating conjunction
  • der Einzug = subject
  • sich = reflexive pronoun
  • verschiebt = conjugated verb, placed at the end

This is a very common German pattern:

• wenn er kommt
• weil ich müde bin
• dass wir bleiben

Because the sentence starts with the subordinate clause:

• Falls der Einzug sich verschiebt, ...

That whole clause takes up the first position in the sentence. In a German main clause, the finite verb must be in second position. So the next part has to begin with the verb:

• ..., bleiben wir für ein paar Tage bei meiner Mutter.

This is called the V2 rule (verb-second).

Compare:

• Wir bleiben bei meiner Mutter.
• Falls der Einzug sich verschiebt, bleiben wir bei meiner Mutter.

You do not say:

• Falls der Einzug sich verschiebt, wir bleiben ... ❌

Der Einzug is a noun meaning the move-in, the act of moving in, or sometimes the move-in date/process.

It comes from the verb einziehen, which can mean to move in.

So der Einzug is not just moving in a general sense. It specifically refers to moving into a new home or apartment.

Examples:

• Der Einzug ist am Freitag. = The move-in is on Friday.
• Wegen des Einzugs haben wir viel Stress. = Because of the move-in, we have a lot of stress.

Here, yes: sich verschieben means to be postponed, to get delayed, or to shift.

German often uses a reflexive form where English uses a passive or a different structure.

So:

• Der Einzug verschiebt sich. = The move-in gets postponed / is delayed.

Compare the two patterns:

• etwas verschieben = to postpone/move something
  • Wir verschieben den Einzug.
• sich verschieben = to get postponed / shift
  • Der Einzug verschiebt sich.

In your sentence, sich refers back to der Einzug.

Because der Einzug is the subject of the subordinate clause, so it is in the nominative case.

The clause is:

• der Einzug = subject
• sich verschiebt = verb phrase

The noun Einzug is masculine, so its nominative singular article is der.

Case forms:

• der Einzug = nominative
• den Einzug = accusative
• dem Einzug = dative

Here, since the move-in is the thing that gets delayed, it is the subject, so der Einzug is correct.

Because German uses a comma to separate a subordinate clause from the main clause.

So in:

• Falls der Einzug sich verschiebt, bleiben wir ...

the part starting with falls is a subordinate clause, and it must be separated by a comma.

This is much stricter in German than in English. In German, these commas are generally required, not optional.

Für means for, and here it introduces a duration of time:

• für ein paar Tage = for a few days

A few useful points:

• für takes the accusative
• ein paar means a few or a couple of
• Tage is the plural of Tag

So this is just a standard way to express duration.

Similar examples:

• für eine Woche = for a week
• für zwei Monate = for two months
• für ein paar Stunden = for a few hours

Because bei is used for staying at someone's place or being with someone.

So:

• bei meiner Mutter bleiben = to stay with / at my mother’s place

By contrast, zu meiner Mutter usually suggests movement toward her:

• Ich gehe zu meiner Mutter. = I’m going to my mother’s place.

In your sentence, the idea is not movement, but location/staying, so bei is the natural choice.

Also, bei takes the dative, which is why you get:

• meiner Mutter (dative)

Because bei always takes the dative case.

The base form is:

• meine Mutter = nominative/accusative

But after bei, it becomes dative:

• bei meiner Mutter

Since Mutter is feminine, the possessive mein- changes to meiner in the dative singular.

So:

• meine Mutter = my mother
• bei meiner Mutter = at/with my mother’s place

This same pattern appears with other dative prepositions too:

• mit meiner Mutter
• von meiner Mutter
• aus meiner Mutter ❌ (not meaningful here, but same dative pattern grammatically)