The concept of "the continuation of movement" applies to all partnership dancing. The partnership is in motion all of the time in one form or another.
American Style Tango is described as the dance of "stops". In reality there is always a part of the partnership in motion. In a forward step, one leg and the body moves and stops on the first beat of the Slow but the trailing leg continues to close on the second beat and on into the next action. So in reality there is a part of the partnership moving all of the time.
Another example would be rise and fall in Waltz. It can be compared to throwing a ball straight up in the air. When you are outside the ball rises to a point where it gently changes direction and starts to fall. When you are in a building the ball hits the ceiling (stops) and suddenly bounces back down. You want Waltz to feel like the ball thrown outside. I always visualize that I only travel horizontally on the one beat and I am moving up and down (rising & falling) on the two and three beats. The two and three beats are just used to get ready to do another one beat horizontal step. The movement horizontally or vertically is "the continuation of movement". The movement does not stop, it just goes in a different direction.
When people first start to dance, they tend to stop completely at the end of each pattern, but as they get more accomplished the movement never stops.
From the ladies (followers) point of view, when a movement is started it continues until the man (leader) ends it. One of the most common errors made by ladies is ending their movement before the leader does. This can cause the partnership to completely stop instead of having the feeling of continuation. An example would be a "walk around turn" in Rumba. Normally the man would initiate the turn by releasing the ladies right hand from a "crossover break" and stop the turn by returning to "closed dance position". However, the man can continue to rotate and pick up the ladies left hand with his right hand into a "back crossover break". If the lady stops prematurely, thinking she is going back to "closed dance position" the "back crossover break" can't be accomplished.
From the man's (leaders) point of view, he must allow the lady to have "the continuation of movement". For example, it is very common for the man to lead the return of a break step (rock step) before the lady has had a chance to finish it. When lead properly, the man will feel a rebound at the end of the break step signaling that the lady has completed her movement and that he should lead the return.
West Coast Swing is very dependent on "the continuation of movement" from the leader and the follower to develop the dance's unique feeling. The continuous connection required in W.C. Swing is only there when you have "the continuation of movement". If the connection is not there the follower is just going through the motions of the pattern and is not really dancing it.
"The continuation of movement" requires patience by both the leader and the follower. Dancing too fast is the main obstacle to acquiring it. Typically the leader looks too far ahead in the pattern and rushes by the opportunity to allow "the continuation of movement". The leader must dance every step to completion before going on to the next. Take your time and enjoy every movement.
"The continuation of movement" is an element that can help you progress from a GOOD dancer to GREAT dancer.