Mahasi Sayadaw Walking

... I am wondering where it can be found that the Buddha taught ...

Walking is considered an antidote to drowsiness and sleepiness in the spirit of the following Sutta, when practiced in combination with sitting practice.

... Bhikshus, what more should be done? Bhikshus, you should train yourselves thus, We will devote ourselves to wakefulness. During the day, while walking to and fro and sitting down, we will purify the mind of obstructions. During the first watch of the night, while walking to and fro and sitting down, we will purify the mind of obstructions (avarana synonym for nīvarana). During the middle watch of the night, we will, after mentally noting the time for rising, mindfully and fully aware lie down, lion-like on our right side, one foot placed on the other. ...

Source: Maha Assa,pura Sutta. Also see footnote 30. The context here is staying awake and fighting drowsiness.

and

If, Moggallāna, that drowsiness still would not go away, then, Moggallāna, you should, perceiving before and after, be resolute in walking back and for, with the senses turned inward, with the mind not straying outward. **It is possible that when you do so, that drowsiness would go away.

Source: Pacalā Sutta. The context here is staying awake and fighting drowsiness.

These suttas only advocate this as an antidote to sleepiness and as a means to keep awake than any other form of pratice.

Another enemy is laziness, drowsiness. All night you slept soundly, and yet when you sit to meditate, you feel very sleepy. This sleepiness is caused by your mental impurities, which would be driven out by the practice of Vipassana, and which therefore try to stop you from meditating. You must fight to prevent this enemy from overpowering you. Breathe slightly hard, or else get up, sprinkle cold water on your eyes, or walk a little, and then sit again.

Source: The Discourse Summaries by S.N.Goenka

... or how the venerable sir came to this instruction.

Perhaps this is a preventive measure to avoid sleepiness during meditation with both being in equal proportions there is less probability of this happening than being the Buddha's instructions. Or perhaps this is what he found successful in the way he practiced hence taught this way. If you find this suitable for your pratice then suitable for you then pratice it as some may find it helpful, though this may vary from individual to individual.

I believe that some foundation for this way of practicing can be derived from the "Five Spiritual Faculties (when fully developed they are called the Five Powers)". These are briefly:

• Mindfulness (sati)
• Energy (viriya)
• Concentration (samadhi)
• Faith (saddha)
• Wisdom (paññā)

These five faculties must be balanced in order for one to progress on the path.

A common example which Mahasi Sayadaw addresses in his book "Satipatthana Vipassana", p. 11, is the 3rd hindrance, i.e. Sloth and torpor also called Dulness and drowsiness or short Laziness.

The Sayadaw explains that laziness happens when there is an excess of concentration and too little energy to fuel or drive that concentration. On the other hand if one has too little concentration and too much energy there will arise restlessness.

So these faculties need to be balanced and one way of doing that is to balance sitting with walking meditation. If one's energy faculty is low but concentration high then one needs to arouse energy. There are several ways of doing that. One way is to do walking meditation in a brisk tempo before doing sitting meditation.

So to partly answer your question. I think the answer could be based on the need for balancing the five spiritual faculties.

Lastly, here is a great dhamma talk by Ajahn Punnadhammo (teaches the Mahasi-method) about how to balance the spiritual faculties.

(The website is down at the moment, I will add the link when it becomes available).