There are two passages which I think answer your question.
““Do not think that I came to bring peace on earth. I did not come to bring peace but a sword. For I have come to ‘set a man against his father, a daughter against her mother, and a daughter-in-law against her mother-in-law’; and ‘a man’s enemies will be those of his own household.’ He who loves father or mother more than Me is not worthy of Me. And he who loves son or daughter more than Me is not worthy of Me. And he who does not take his cross and follow after Me is not worthy of Me. He who finds his life will lose it, and he who loses his life for My sake will find it.” Matthew 10:34-39 NKJV
Here, Jesus is saying that He must have first priority in our life. That we must love Him more than our spouse, or child, or parents.
The second passage comes from the same chapter:
““Behold, I send you out as sheep in the midst of wolves. Therefore be wise as serpents and harmless as doves. But beware of men, for they will deliver you up to councils and scourge you in their synagogues. You will be brought before governors and kings for My sake, as a testimony to them and to the Gentiles. But when they deliver you up, do not worry about how or what you should speak. For it will be given to you in that hour what you should speak; for it is not you who speak, but the Spirit of your Father who speaks in you. “Now brother will deliver up brother to death, and a father his child; and children will rise up against parents and cause them to be put to death. And you will be hated by all for My name’s sake. But he who endures to the end will be saved. When they persecute you in this city, flee to another. For assuredly, I say to you, you will not have gone through the cities of Israel before the Son of Man comes. “A disciple is not above his teacher, nor a servant above his master. It is enough for a disciple that he be like his teacher, and a servant like his master. If they have called the master of the house Beelzebub, how much more will they call those of his household! Therefore do not fear them. For there is nothing covered that will not be revealed, and hidden that will not be known.” Matthew 10:16-26 NKJV
Jesus made it clear that persecution would come to His followers and it has from the first disciples down through the ages to us. Uniquely in Jesus day, Jesus let the disciple know that the religious (the “devout” Jews) will think they are doing a service to God to get rid of you. There may be applications of this today where “religious” groups think they are doing the will of God in killing and persecuting Christians.
Although a time will come, and has come for many persecuted Christian, where one’s own family betrays you because of your faith in Christ, yet Jesus encourages us. We will be hated, me may be put to death, we have need of endurance perhaps unto death but we will be saved in the end by the blood of our Savior for all eternity. If persecution strikes your city, Jesus instructs us to flee. It is ok to run somewhere else unless the Lord uniquely calls you to stay according to His will as I believe we see in other places in the Scriptures, particularly at times in the life of Paul and the Prophets. But Jesus in this passage does not say anything against running when persecuted. He even says if you run when persecuted (which multiple times has resulted in the blessing of the Gospel being spread to new places) His followers would not make it through all the cities of the small nation of Israel before He returns. The emphasis is on the nearness and blessed hope of His soon return.
We, Believers, are not above our Master Jesus. If He was called a demon-possessed troublemaker, hated by His contemporaries, we as His followers will not escape identifying with Him today and sharing in His sufferings. But notice the last part of this passage! We are not to fear for the Lord will bring things to light, and what is secret to knowledge one day. Brothers and sisters, it is not easy to follow Jesus but it is worth it. He is worthy!