I looked in M-W, but none of the meanings of the English verb swing seem to describe what kids do when they sit on a swing. Meaning 3 c of the intransitive verb comes close – "to convey oneself by grasping a fixed support – swing aboard the train" –, but if swing is the right word, I would have expected something more explicit.
For example, in German, the first meaning given in the most widely used dictionary for the verb schaukeln is:
1.a. (auf einer Schaukel o. Ä.) auf und ab, vor und zurück, hin und her schwingen [swinging up and down, back and forth, or sidedways (on a swing or similar)]
But German schaukeln is not equivalent to English swing. The German word that covers most of the meanings of the English verb swing is the etymologically related verb schwingen. So in German, schwingen "swing" is not the verb for moving back and forth on a swing. But if one hears the word schaukeln in German, the first thought that pops into ones mind mind is someone sitting on a swing.
Now schaukeln is usually translated into English as rock, which doesn't seem to be the right word for what one does on a swing either. Neither rock nor swing come with examples that have anything to do with a playground.
The meanings appear to me like this:
English German swing = schwingen <- "hanging from something and swinging at its end" rock = schaukeln <- "moving back and forth, up and down, or sideways" ? = schaukeln <- "sitting and moving on a swing or seesaw"
So is there another verb that is more commonly used for this activity in English?