It is quite possible within certain parameters. Those parameters would include the ability of the camera body to regulate the continuous drive rate, the ability of the flash to provide the needed light at a power level setting that allows for a fast recycle rate, and the ability of the photographer to find the rate where the capabilities of the camera and flash can work together for the best combined results.
If you can combine the ability to control your burst rate to something around 3 or 4 frames per second and can light the scene adequately with a very low power setting, with many speedlight models it is possible to have the flash fire for every frame in such a burst. You just need to play around with the components you are using and see where your "sweet spot" is.
Some cameras will allow the user to set a continuous drive mode rate that is slower than the maximum for the camera. I've had cameras that gave a choice between, say, 8fps high speed continuous or 3 fps low speed continuous. I also have a camera that will allow me to set the rate from between 1 fps and 8 fps for the low speed continuous setting and from between 2 fps and 10 fps for the high speed continuous setting. There are also quiet modes on some cameras that will reduce the maximum frame rate.
How long a flash takes to recharge after a particular discharge is dependent upon the total capacity of the flash's capacitors and the amount of power used for a particular strobe. Most speedlights control flash power by controlling the duration of the flash. When set to very low power only a fraction of the flash's total capacity is actually discharged. This allows the capacitors to be raised back to full capacity much faster than if a full power discharge had been used. If the flash is capable of drawing power from an external battery pack this will also usually speed up the time needed to recycle.
This is what the external battery pack port looks like on your YN560 II flashes:
It's on the left (That's the PC Sync connector on the right). It is a fairly standard pin arrangement that most generic external battery packs will fit.
A few flashes will allow the flash to refire as soon as the shutter is pressed with whatever amount of energy is stored in the capacitors. Most won't fire, though, until at least the amount of energy needed for the power to which the flash is set is stored in the capacitors. Some won't fire until the capacitors are fully charged, even if power is set to a low level.