For purposes of Branching Evolution, "mov[ing]" counters from one creature onto another includes "put[ting]" those counters onto the latter creature (C.R. 122.5), so that Branching Evolution would double the number of +1/+1 counters put on the latter creature this way.
In general, a counter is "put" on a permanent if the effect says "put ... counter[s]", "move ... counter[s]" (C.R. 122.5), "double the number of ... counter[s]" (C.R. 701.9e), or "enter[s]" or "put ... onto" or "return ... to" "the battlefield ... with ... counter[s]" (C.R. 122.6). See also this thread and this thread.
Note that Branching Evolution's ability is not a triggered ability (it lacks "when", "whenever", and "at") (C.R. 603.1).
For purposes of Branching Evolution, if you "double the number of +1/+1 counters" on an object, you normally put as many +1/+1 counters on it as it has, so that with Branching Evolution, you put twice that many of those counters instead (C.R. 701.9e). For example, if you control Branching Evolution, Kalonian Hydra has three +1/+1 counters on it, and you "double the number of +1/+1 counters" on it, you put six more +1/+1 counters on it (rather than three) for a total of nine +1/+1 counters.
Note again that Branching Evolution's ability is not a triggered ability, so that ability doesn't "trigger".
As far as the rules are concerned, putting counters on an object includes—
"putting counters on that object while it's on the battlefield", regardless of what verb is used to refer to that action (whether "put", "move" [C.R. 122.5], "double" [C.R. 701.9e], "give", or some other verb), and
the case of "an object that's given counters as it enters the battlefield" (e.g., "enters the battlefield with ... counters")