Kayasankara is the breath (MN 44).
Kayasankhara is properly translated as ‘body conditioner’, similar to the meaning of ‘body sustainer’ or ‘body nourisher’.
The 3rd step of Anapanasati is properly translated as 'experiencing all bodies'. There are three 'bodies' or 'kaya', namely, breath body, physical body (rupa-kaya) and mental body (nama-kaya). In the Anapanasati Sutta, the Buddha says: "the breath is a body among other bodies". In brief, the 3rd step is to experienced how the three bodies interrelate with or inter-condition each other. For example, when the mind is calm, the breath will be calm, then the body will be calm; if the mind is agitated, the breath will be agitated; then the body will be agitated.
The 4th step of Anapanasati is training how to use the mental body (mind) to tranquilize the breathing body; which in turn tranquilizes the physical body; which in turn tranquilizes the mind and eventually makes the mind feel happy (step 5 & 6, rapture & happiness).
In summary, tranquilizing the kayasankhara means tranquilizing the breathing, which will simultaneously tranquilize the physical body.
Refer to Bhikkhu Buddhadasa's book: Mindfulness With Breathing - Unveiling the Secrets of Life:
a Manual for Serious Beginners.