Jesus pretty much stayed in the boundaries of Israel, and even so, Israel hosted many travelers from other countries and neighbouring regions so that His fame went well past the physical boundaries of travel.
When He did travel to Tyre and Sidon, He did so for the direct purpose of finding rest, a place where they did not know Him. Unfortunately that rest did not come. The mother of a demoniac daughter was hoping that Jesus would come to her home and He had.
When she approached Jesus, He did not insult her nor was He insensitive in His rejection. He was both sensitive to her need as well as His desire to teach something to His disciples. He used well known household imagery in which to create an opportunity to lead the woman to a maturity in her faith, a more precise hope. He also was wanting the disciples to see their role and their privilege in ministering to a needy, faith-filled heart. For both her and the disciples He showed that the greater the test comes greater faith in God.
In asking for mercy for herself and healing for her daughter to the “son of David” she knew of the Jewish hope in the Messiah, as did the woman at Jacob’s well. She hoped for, and expected to share in that Messianic blessing.
Just then a Canaanite woman from that region came out and started shouting, “Have mercy on me, Lord, Son of David; my daughter is tormented by a demon.” But he did not answer her at all. And his disciples came and urged him, saying, “Send her away, for she keeps shouting after us.” He answered, “I was sent only to the lost sheep of the house of Israel.” But she came and knelt before him, saying, “Lord, help me.” He answered, “It is not fair to take the children’s food and throw it to the dogs.” She said, “Yes, Lord, yet even the dogs eat the crumbs that fall from their masters’ table.” Then Jesus answered her, “Woman, great is your faith! Let it be done for you as you wish.” And her daughter was healed instantly. – Matthew 15:22-28