I had forgotten how they worked.
Yesterday I met my downstairs neighbor by the stairwell. We chatted about a few things. She was nice enough to give me an extra flyswatter, as we'd both had recurring problems with the little things and she'd bought two.
Today I saw a housefly hovering in my bathroom. It flew up to the window over the shower and I swatted at it. Since I hadn't made contact with it, I assumed that it had escaped. But a second later, its body fell into the tub.
I don't enjoy killing flies, and I wish them a peaceful life wherever their natural environment is. However, neither am I going to let it pass as they invade my space, and this is at least a less stressful way of getting rid of them.