See this short article by Sayādaw U Sīlānanda — The Benefits of Walking Meditation
Read what Sayādaw U Pandita says about Walking Meditation
Walking meditation develops balance and accuracy of awareness as well as durability of concentration. One can observe very profound aspects of the Dhamma while walking, and even get enlightened! In fact a yogi who does not do walking meditation before sitting is like a car with a rundown battery. He or she will have a difficult time starting the engine of mindfulness when sitting.
Ajahn Chah used to look at the walking path outside his monk's kutis — if they were not well worn he knew they were being negligent in their practise. Ajahn Mun was also a diligent practitioner of walking meditation.
In my personal experience, the best insights came from walking, not sitting. In Burma, its very hot and humid. Sitting for long hours is tough. Walking continuously for long periods is not difficult. Continuous mindfulness leads to insights.
Famous examples from the Buddha's time are Venerable Ānanda who gained Arahantship after striving in walking practice for the whole night. Realising that he needed to balance the faculties, he went to lay down on his couch. Between sitting down and before his head touched the pillow, he passed through all three stages of the Path — Once-Returner, Non-Returner, and Arahant.
Venerable Moggallāna also practised walking relentlessly from the time of his ordination, until finally gaining Arahantship while sitting.