I did some more research on the usage of διώκω, and here’s what I came up with.
The verb διώκω mainly means to chase or put to flight, and it’s related to δίεμαι. The less common active of δίεμαι is δίω, which is to flee or be afraid. There is also καταδιώκω. The metaphorical or specialized usage to mean “hound, persecute” exists in classical Greek, but it’s rare.
In the septuagint (early koine), it’s mainly used to translate the Hebrew rdf (רָדַף, chase), although it also occurs as a translation of about a dozen other Hebrew verbs such as dhd (rush, dash), shdd (ravage, despoil), rdh (rule), shft (judge, govern), and chrd (tremble, be afraid).
Most usages in the septuagint mean simply to chase. Examples (WEB): “Barak pursued the chariots” (Judges 4), “he who chases fantasies is void of understanding” (Proverbs 12:11), “all who pursue the word of the god of Israel” (2 Es?). A slogan that recurs in varying forms is “How could one chase a thousand, and two put ten thousand to flight” (Deuteronomy 32:30, in a song sung by Moses as he presents the law to the people).
It’s very rare in the septuagint to find the verb meaning “persecute.” Out of about 80 usages, only one is clearly what we would mean by “persecute:” “Why do you persecute me as God, and are not satisfied with my flesh?” (Job 19:22). There are a few other possible semantic matches, such as Deuteronomy 30:7, Lamentations 1:3, and 1 Maccabees 2:47, but these are less clear-cut.
The semantics of the verb seem to be such that it can be: positive (seeking knowledge); natural or impersonal (the sun chasing the sea foam); or meaning to drive or propel (a ship being driven by the wind). It is never: instantaneous; symmetrical or used to describe strife; done from in front; an action against someone already at hand, bound, or confined. These considerations suggest that usages such as those in the sermon on the mount probably could not be understood by Greek speakers as referring to oppression (Romans against conquered lands, second temple authorities against Galileans) or sectarian strife.
Based on all of this, I think it’s a little difficult to have Matthew 5:10, 5:11, 5:12, or 5:44 be accurate translations of an actual Aramaic speech given by Jesus to his followers in Palestine, and meant to be intelligible to them. The repeated and heavy emphasis on the same Greek verb would have to be a translation of a single verb in an Aramaic original, and only rdf seems to make sense – the other possibilities (run, ravage, rule, cause fear, et al.) don’t seem to fit the context.
The best hint that Jesus gives as to his meaning is “for that is how they persecuted the prophets who were before you” (Matt 5:12).
If this is meant to warn of the rounding up of the movement’s leaders, then that would make sense in terms of the historical record. At least according to the gospels’ chronology, John the Baptist could already have been killed by the time of this speech, and Josephus also documents the execution of both Jesus and his brother James. But this is all about the leadership, and it doesn’t seem reasonable for Jesus’s listeners to make the leap to imagining themselves as victims or possible future victims of persecution. (There have been suggestions that some portions of the sermon on the mount are addressed to the crowd, while others are to the twelve.)
The extended discourse on persecution could work as a prediction of a generalized persecution of the movement’s followers, but no such persecution existed until around 250 CE, and even if we put aside the question of a naturalistic or supernatural method by which Jesus could make such a prediction, such a prediction would not be understandable (or useful) to his listeners in 30 CE.
Subject to the various uncertainties in this analysis, it seems to me like it really points toward the idea that at least this portion of the sermon on the mount (most or all of Matthew 5?) has to be the product of a later era of the early church, during which the execution of people like Paul and James led to a self-definition of the emerging religion as a persecuted group. This late dating of the material would also fit well with the fact that the usage of διώκω to mean “persecute” seems to have been quite rare until the Pauline school and other early church groups began using it frequently that way. A late dating was suggested by Bultmann.