You will find two main cases when this is an issue. You either have a small portion of your image with movement, such as humans or vehicles. Or you have a large portion of the image with movement such as when photographing from a moving vehicle.
Small Subjects With Movement
This works just fine when the image contains some moving objects but is not mainly composed of moving objects.
In the example at the bottom, I had many moving objects(cars), but the majority of the image is still comprised of static buildings. The first image is right after creating the HDR/tonemapped image. The second is after my "fix". My fix was bringing the tonemapped image into Photoshop, bringing in the middle or "properly exposed" image as a layer, and masking in or out the portions that I wanted.
Essentially in the final image you only see one version of the cars because that is all that is visible, a single frame and not each "ghost" or partial capture of a moving car that the HDR had in it.
It can take a bit of playing in Photoshop to get this right, and sometimes to get the moving object to look good, you need to adjust levels of the image you are masking back in to match the new levels of the tonemapped image.
Majority of Subject Has Movement
This technique doesn't obviously work great for an image comprised primarily of a moving object. In that case, the best way that I have found is to use the highest shutter speed possible, and let the HDR software auto align the images. It will attempt to correct for the slight movement between frames on its own, and sometimes this works out.
The above is only really possible when you have a camera that can auto-bracket, otherwise doing so manually using dials or controls on the camera slows down the process enough that it isn't really possible.
As you pointed out in the question, another option is to create a single image HDR. This is kind of a "way out" if you ask me, as it doesn't really give you much more detail and is not worth it in many cases. But if you have no other choice this might be something you can try, but your results aren't going to be as dramatic as a multi-image HDR.
Image after HDR creation and Tonemapping
Image after HDR/tonemapping and Photoshop masking