Abarim Publications' online Biblical Greek Dictionary
The preposition κατα (kata) expresses a motion downward. Tied to appropriate verbs it usually describes motion down from a higher place or down to a lower place. It may be used in the sense of "bearing down upon", that is an aiming for a certain place or even a certain understanding, when something "comes down" to something. Our word may also refer to something in the sense of "in respect of" or "on account of" or "by means of".
Our preposition may also be used to describe an opposite motion (in whichever direction), a counter motion or counter force. It may reflect hostility toward someone in the sense of warning or accusing, or direct humiliating or putting down of someone.
In the Greek mind, traversing a place was expressed as going down it, and as such our preposition may be used to mean through or over (as in English: traveling down a road). It may also be used to simply emphasize the being at a location (as in English: the church that is down at her house — Romans 16:5), or even some particular point in time ("down" on Tuesday we eat beans).
Our preposition κατα (kata) may be used to emphasize pervasiveness in the sense of all throughout (down to the last spot) or totality. It may be used to establish a comparative (according to the flesh) or compliance (according to the law).
When the core verb already describes a motion downward, our preposition κατα (kata) may be added as an intensifier.