Russian Word Formation - The Verb of Motion

Note the table labeled: Prepositions Expressing Basic Motions
And the list of prefixes


к treats its object like more of a standing(vertical) unit with presence (most commonly it's a person), whereas в is something with an interior, and на is something with a surface generally (neither usually a person).
к перекрёстку gives the idea of "up to" the edges of the crossroads as opposed to into it.

Also, the list of prefixes explains how certain verbs will correspond to certain prepositions.