The violent imagery seems to be due to dependence on Micah. Note that Matthew 10:34-39, wherein Jesus says that he has come to lay a sword on the earth, paraphrases a non-LXX Greek version of Micah 7:5-6, of "daughter rising up against her mother, the daughter-in-law against her mother-in-law," and so forth. As early as Tertullian was it recognized that this gospel passage refers to a prediction of Micah (Contra Maricon, 4.29). As for Luke 16:16=Matthew 11:12, and particularly the use of biastai "violent ones," this passage is possibly dependent on Micah 2:13:
"The one who opens the breach (diakopes) will go up before them. They [the sheep] will break through (diekopsan) and pass through the gate and go out by it. So their king (basileus) will go before them, and the Lord (kurios) at their head". (LXX)
Although the verb is not biazo or biastai "violent ones" in the LXX, interestingly these words render the Hebrew word prts "break through" in Micah in other contexts in the LXX (cf. 2 Samuel 13:25, 27; 2 Kings 5:23), suggesting that the author was dependent on a non-LXX version of Micah that possibly used the word biazo in this passage. In the exegesis of the verse in Q, the one "opening the breach" into the kingdom would have been John the Baptist, the sheep "breaking through" into the kingdom would correspond to his followers, and the "king" and "Lord" would represent Jesus who goes ahead of the sheep and leading them into the kingdom. In the parallel of Luke 16:16, the phrase "everyone enters it [the kingdom] violently" does not seem to assume those attacking the kingdom but those who are striving with forceful effort (like the sheep struggling to enter the breach) to enter it. The thought thus seems close to Micah 2:13.
However, the use of the verb harpazousin "seize" later in the verse suggests a further image: that of soldiers breaking through a city's outer defenses and plundering the treasures within. In fact, this is exactly the interpretation in the Gospel of the Nazoreans which renders this final clause as "the kingdom of heaven is plundered" (cf. the Zion Gospel). The Hebrew word that is equivalent to harpazousin, namely 'chztm, interestingly appears throughout the OT as referring to the Israelites' seizure of Canaan as their inheritance (cf. Genesis 17:8, 48:4; Leviticus 25:34; Joshua 22:9), raising the possibility that the metaphor in the passage is that of Joshua (=Jesus) leading the Israelites (= Jesus' followers) into the kingdom, forcibly entering it with war as the Israelites fought their way into the Promised Land. The Hebrew verb prts in Micah 2:13 is also the basis of the name of Perez, son of Judah, in Genesis 38:29, who in the geneology of Matthew 1:3 was the ancestor of Jesus.