Die Kinder kommen mit Schlamm an den Schuhen nach Hause.

German often uses kommen when the movement is seen toward the destination or from the speaker’s point of view as an arrival.

So:

• nach Hause kommen = to come home
• nach Hause gehen = to go home

Both can be correct in different contexts, but kommen focuses more on the children arriving home.

This is a very common German distinction:

• nach Hause = homeward / to home → direction, movement
• zu Hause = at home → location, no movement

So:

• Die Kinder kommen nach Hause. = The children come home.
• Die Kinder sind zu Hause. = The children are at home.

In your sentence, the children are moving, so nach Hause is the correct form.

German usually uses an in expressions like this when something is attached to or stuck to a body part, piece of clothing, or object.

So:

• Schlamm an den Schuhen = mud on the shoes
• Blut an der Hand = blood on the hand
• Farbe an der Jacke = paint on the jacket

auf often suggests something resting on top of a surface, while an fits better for something clinging to or showing on the shoes.

In this sentence, the mud is stuck to the shoes, so an den Schuhen sounds natural.

It is dative plural.

Here’s why:

• an can take either accusative or dative
• with location (no movement), it takes dative
• the mud is located on the shoes, so this is a location idea

So:

• an den Schuhen = on the shoes

Also, in the dative plural, German usually adds -n to the noun if possible:

• die Schuhe → den Schuhen

German often leaves out possessive words like their, his, or her when the meaning is obvious from the context.

So instead of saying:

• mit Schlamm an ihren Schuhen

German can simply say:

• mit Schlamm an den Schuhen

Because the subject is die Kinder, it is already clear whose shoes are meant: the children’s shoes.

Both versions are possible, but the version without ihren is very natural.

Because Schlamm here is being used in a general, uncountable sense: just mud, not a specific mud already identified.

Compare:

• mit Schlamm = with mud
• mit dem Schlamm = with the mud

In this sentence, the meaning is general: the children have mud on their shoes. So mit Schlamm is the natural choice.

It is a prepositional phrase describing the condition in which the children come home.

Literally, it is something like:

• with mud on the shoes

The structure is:

• mit Schlamm = with mud
• an den Schuhen = on the shoes

Together, it describes the children when they arrive home.

A very natural English translation is:

• The children come home with mud on their shoes.

It is most naturally understood as connected to Schlamm.

So the phrase groups like this:

• mit [Schlamm an den Schuhen]

That means:

• with mud on the shoes

It is not saying that the children come home at the shoes. Instead, an den Schuhen tells you where the mud is.

Because German main clauses normally follow the verb-second rule.

In this sentence:

• Die Kinder = position 1
• kommen = position 2

Everything else comes after that:

• mit Schlamm an den Schuhen
• nach Hause

So the basic pattern is very normal German word order.

Yes, German word order is somewhat flexible, especially with adverbials and prepositional phrases.

For example, you could also say:

• Die Kinder kommen nach Hause mit Schlamm an den Schuhen.

But Die Kinder kommen mit Schlamm an den Schuhen nach Hause sounds smooth and natural because the descriptive phrase comes before the destination.

You could also front another element for emphasis:

• Mit Schlamm an den Schuhen kommen die Kinder nach Hause.

That puts extra focus on the muddy shoes.

No. Even though you see an in the sentence, this is not the separable verb ankommen.

Here, the parts are:

• kommen = come
• an den Schuhen = on the shoes

So an belongs to the prepositional phrase an den Schuhen, not to the verb.

If it were ankommen, it would mean arrive:

• Die Kinder kommen nach Hause an.
  This is not natural here, because nach Hause kommen already expresses the idea clearly.

German and English handle body parts and clothing a little differently.

German often uses:

• the definite article: den Schuhen
• instead of a possessive: ihren Schuhen

Because the subject already tells us who is involved, German does not always need to repeat their.

Also, Schuhen is plural because the children have mud on their shoes in general, not just one shoe.

Yes, but it changes the nuance.

• mit Schlamm an den Schuhen specifically says the mud is on their shoes
• schlammig means muddy, which is broader

So:

• Die Kinder kommen mit Schlamm an den Schuhen nach Hause.
  = specifically mud on their shoes

• Die Kinder kommen schlammig nach Hause.
  = the children come home muddy

The original sentence is more precise.