Both microtubules and microfilaments move based on the shared feature of motor proteins. In both types the motor proteins have the ability to elongate and/or pull back when needed. In microtubules this protein is tubulin and it can grow by the addition of more tubulin or shrink/even dissipate by the breakdown of tubulin. In microfilaments this protein is actin. It is globular and will "stick" to other filaments to help build additional filaments that act as branches and thus elongates the microfilament.
Work Step by Step
Microtubules-page 114(top of page), Microfilaments-page 115 (bottom of page)