Does speed of body movements affects the mind? Do we have to slow down our actions to have a better sati? It feels it does affects. But are there any sutta about this? How that works?
The ordinary speed of ordinary body movements does not adversely affect the mind. In my experience, the ultra-slow methods of walking meditation do not enhance mindfulness and, in my opinion, hinder mindfulness, because it takes far more unnatural activity of mind to maintain a very slow unnatural walking pace.
The original Buddhist scriptures are silent on the speed of walking meditation, which means walking meditation was most probably done at a natural pace; which is why it is described as "walking back & forth (caṅkama)".
I know from intensive personal practice that natural ordinary body movements are the optimal for mindfulness.
In fact, the what most do not understand is how to practice mindfulness. In higher practise, mindfulness is maintaining the mind is a state of letting go. Therefore, in the state of letting go, body movements are natural rather than contrived and controlling.
In every sati practice, there is an aspect of samatha just as every samatha practice has an aspect of sati. If you're finding that slowing your bodily movements down is helping your sati, it could be an indication that your mind isn't composed enough for effective sati work. In cases like this, I'd recommend that you take some time to still your mind using any of the samatha practices outlined by the Buddha, or, at the very least, slowing down your movements as you have been until sati can be maintained at the normal speed of life. Ideally, the efficacy of your sati shouldn't be determined by where you are nor be dependent on what you are doing.
It's not that fast body movements reduce the calmness of the mind.
Rather, it's the mind that has to be agitated to generate deliberate body movements which are fast.
From Dhammapada 1:
Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with an impure mind a person speaks or acts suffering follows him like the wheel that follows the foot of the ox.
Mind precedes all mental states. Mind is their chief; they are all mind-wrought. If with a pure mind a person speaks or acts happiness follows him like his never-departing shadow.