mano a mano; mano mano; man mano
As far as Italians are concerned these
expressions (freely exchangeable) have only one use and meaning, the origin of which - most likely - has to do with the way in old times people used to measure short distances by using an open hand.
You put a hand widely open beside another end moving on as you take measures. From this habit in old times may have arisen the idea of following an event - shal we say - "step-by-step" (if you're measuring long distances) ... or similarly "hand-by-hand" (if you're measuring, say, a cloth or a stick of wood, etc.).
By extension, every time an Italian wants to mean following something very closely and at every new event uses this
expression.
I mark
on the wall my child's height as (mano a mano) he grows.