The carriage is a symbol of majesty and power, and a chariot also represents the sun and the divine.
The cart is a more mundane symbol signifying down-to-earth hard work.
Psychological / emotional perspective: Any symbol which signifies our being moved in some way draws attention to our ability to make progressive changes in our lives. We may have to explore archetypal images for an understanding of our own motivations.
The chariot in the sense of a gladiatorial vehicle can represent basic urges before they have been honed by conditioning and knowledge. Again a cart is representative of workaday concerns.
Material aspects: In modern times most people will dream about the car or other similar forms of transport.
Dreaming of a carriage, such as a horse-drawn one, could be suggestive of old-fashioned attitudes to modern thinking, and a chariot means we have to accept a very basic kind of control.
A train carriage would indicate that we are taking a ‘journey’ that is slightly more public than a car journey.