Do teleconverters work properly for recording movies with hybrid cameras?
It really depends on the capabilities of the focus system on the camera and the particular teleconverter and lens used.  Since you mention action videos, I assume you will be dependent on autofocus.  Not all teleconverters are able to properly communicate focal length and focus adjustment controls, so you'll need a "smart" teleconverter.  

Additionally, teleconverters have an impact on the f-stop at which a lens can shoot and because of the way they operate, with some lenses, they limit the aperture settings at which autofocus can occur.  You'll want to check the documentation of the particular teleconverter you intend to use to make sure it is compatible with the lens and aperture you wish to shoot at.

Finally, there is the concern of the precision of the AF motors.  Extending the focal length of a lens also narrows the depth of field and makes the precision of the AF motor much more critical to avoiding focus "breathing" where it goes in and then back out of focus.  This is also going to be lens specific and some lenses will do better than others.

This is all running on the assumption that you have hybrid pixel phase detect autofocus on the hybrid camera in question (or some other form of non-contrast based phase detection that allows direct travel to the proper focus point.)  If you do not, then you will have issues with focus breathing with or without the teleconverter that will be less than ideal for video purposes.