The material in the Earth's outer core is both a liquid and an electric conductor. Fluid flow is driven by thermal convection currents. The source of heat driving the convection is the decay of radioactive elements that are also carried by the fluid flow. Fluid flow and electric currents are coupled through electromagnetic forces. The core is also likely of inhomogeneous composition. Charge is not only carried along by the fluid flow, electromagnetic induction can create eddy currents. The equations describing this very complex system cannot be solved without fairly large computers. Numerical models of the Earth's magnetic field indicate that the field is 'chaotic' - spontaneously changing its shape and polarity. Because of this complexity, it is not necessary that the direction of flow to reverse everywhere for the direction of electric currents to change and cause the magnetic field to change too. Because the system is 'chaotic', a relatively small change in flow might cause a large change (even reversal) of the magnetic field. Although the computer models are simulations, they have become very good at reproducing the secular variation of the magnetic field we observe at the Earth's surface.
You would have to get the technical and mathematical details out of the scientific literature.
The previously cited article from National Geographic is a good popular article, the researchers interviewed are still active and you go search for their latest publications.
I don't have a single technical seminal citation to give, but one place you might start researching the literature is the web pages of the Geomagnetism & Paleomagnetism Section of the American Geophysical Union (AGU). You'll find information there for both researchers and students. Look for information on the web at other non-US National surveys as well for terms such as "Geomagnetism" and "Geodynamo."
I still like the 2010 book: "North Pole, South Pole: The Epic Quest to Solve the Great Mystery of Earth's Magnetism" by Gillian Turner. A non-mathematical treatment of the history, written by a geophysicist, but also gives the names of scientists are still actively conducting research, search to see what they and their students may be writing about today.