to wander / digress / roam
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to wander
To let one's gaze or thoughts wander or move around without a fixed goal.
Während der langweiligen Präsentation ließ er seinen Blick aus dem Fenster schweifen.
During the boring presentation, he let his gaze wander out the window.
to digress
To move away from the actual topic in a conversation, text, or lecture; to digress.
Können wir zum Thema zurückkommen? Wir schweifen schon wieder ab.
Can we get back to the topic? We're digressing again.
to roam
(archaic, literary) To move from place to place without a fixed goal; to wander around or roam.
Die Nomaden schweiften mit ihren Herden durch die weite Steppe.
The nomads roamed through the vast steppe with their herds.
You can recognize the meaning of 'schweifen' (to wander/digress/roam) from the object: If it refers to 'Blick' (gaze) or 'Gedanken' (thoughts), it is about wandering around. If it refers to 'Thema' (topic), it is about digressing. If it refers to people or animals, it is about roaming around.
Sein Blick schwiff unruhig durch den Raum.
His gaze wandered restlessly around the room.
In this meaning, 'schweifen' (to wander) is very often used in the fixed construction with the verb 'lassen' (to let): 'den Blick schweifen lassen' (to let one's gaze wander) or 'die Gedanken schweifen lassen' (to let one's thoughts wander).
Am Strand ließ sie ihre Gedanken in die Ferne schweifen.
On the beach, she let her thoughts wander into the distance.
When it is about getting away from a topic, the separable verb 'abschweifen' (to digress) is almost always used today. It is more common and clearer than the simple 'schweifen' (to digress).
Entschuldigung, ich bin vom Thema abgeschweift.
Sorry, I digressed from the topic.
The meaning 'umherziehen' (to roam around) is archaic and today is used almost only in literary or historical texts.
In alten Gedichten liest man von Rittern, die durch die Wälder schweiften.
In old poems, you read about knights who roamed through the forests.
When 'schweifen' (to roam) describes movement from one place to another, the perfect tense is formed with the auxiliary verb 'sein' (to be).
Die Wölfe sind nächtelang durch das Revier geschwiffen.
The wolves roamed through the territory for nights on end.
to hike / migrate / wander
A gaze can wander when it moves slowly from one point to the next. Schweifen more strongly emphasizes that a gaze or one's thoughts roam without a fixed goal. Wandern is therefore appropriate when there is a recognizable sequence, while schweifen fits undirected movement.
Im Museum schweifte ihr Blick ohne festes Ziel durch den ganzen Saal.
In the museum, her gaze wandered aimlessly around the entire room.
Im Museum wanderte ihr Blick langsam von einem Bild zum nächsten.
In the museum, her gaze slowly wandered from one painting to the next.
