Hmmm... I will share my experience and understanding, and then maybe others will chime in with their thoughts as well.
The most basic of motions is the arcing or curve-a-linear, the flexion and extension of the muscles causing a single joint articulation. Recruiting multiple joints to form a kinetic chain facilitates the creation of more complex movements such as circles, infinities, and waves.
Now, just because more sophisticated motions such as waves are formed by more rudimentary movements such as arcs, this does not imply that we should concentrate solely on the fundamentals. Becoming aware of our fuller range of motion and human potential provides us with greater awareness and familiarity with what might have been otherwise "uncomfortable," thus increasing our survivability.
For example, in the early days of NHB events, fighters who were from a primarily striking background were easily neutralized by being placed on their backs. Their lack of familiarity with the ground proved to be on obstacle to their defense, but had they spent only a marginal amount of time training this facet of combat, their chances for success would have increased exponentially. I note Maurice Smith as an example. This is only a singular benefit that incrementally complex motions afford us.
I may have gone off on a tangent, but I will try to bring it back around. We begin with the biomechanics; this is the foundation of our knowledge. Once we have reclaimed our inherit movement patterns, we can begin to explore them in context, such as your "takedown" example. The only difference is that there is no need to exaggerate the movements. Simply apply the same sound, natural, efficient movements to the particular venue that you are engaged in. You will cognitively recognize the connection, and more importantly, you will somatically make the connection.
I would like to hear some other ideas about this as well.