Most people kill flies without a second thought. But they are animals, and they can feel pain. So the responsible alternative is to get flies out of the house without killing them. It probably won't take any longer than chasing them around the house with a flyswatter. And if compassion for insects is beyond you, do it for practical reasons; no bloody fly corpses to clean up.
Open a window and coral them out of the house. During the daylight hours, flies tend to congregate on windows anyway trying, in vain, to find a way out. Find a popular window and remove the screen from the outside, open the window from the inside, then watch and wait from a distance. Within five minutes, the flies should be free. Just be sure to close the window right away or you could end up with more flies inside the house than when you started.
Wait for each fly to land on a flat surface, like a window, wall or table. Carefully place a glass on top of it, then slide a thick piece of paper underneath (like a postcard or cardstock), between the glass and the surface. Hold the paper against the glass, carry the fly out of the house and release.
Catch the flies with your bare hands, if you're looking for challenge and don't mind touching them. Yes, it's next to impossible for most, but it can be done, especially with practice.
Invest in a bug vacuum; yes, they do exist! Intended for kids to catch and study bugs without killing them, it works for this adult purpose too. You should be able to get one for around fifteen bucks. But beware. There are bug vacuums that do kill the insects in the process. To find the kind you're looking for, search the Internet for "bug vacuum for kids."