They sure do! Think of our neighborhood in the Milky Way as a giant roadway that encircles the galactic center. The stars are cruising at speeds of a half a million miles per hour. Like an ordinary highway filled with traffic, you sit in one of these cars and see your neighbors moving near you. Some are pulling ahead, some are falling behind as they try to maintain an average speed. Unlike our roadways here on Earth, the Sun and its neighbors can continue circulating along this cosmic beltway for billions of years without a single fender-bender. Near the Sun, the typical relative speeds can be about 10,000 miles per hour, and interlopers from the halo regions of the Milky Way can travel a dazzling 200,000 miles per hour. Our Sun is dragging our solar system in the direction of Vega at a speed of about 43,000 miles per hour. If Vega obliged us and stayed put at its current distance of 26 light years, we would catch up to it in, oh, about a million years. One of the fastest known stars in terms of its local speed compared to the Sun is the recently discovered neutron star RX J185635-3754 located 200 light years from the Sun. It is a 'runaway' from a supernova explosion over one million years ago, and is traveling through space at nearly 240,000 miles per hour. It will reach its minimum distance to Earth of 170 light years, in about 300,000 years.
