When I first read the title to this thread I wanted to respond "Of course, protons move! I just moved a lot of them when I picked up this laptop computer."
But I see you are asking a serious question. You are asking specifically about "charge" moving in an electric line, right? "Charge" is not the same as "protons" or "electrons". If the electrons move in one direction, then a negative "charge" is moving in that direction while a positive "charge" is moving in the opposite direction, even though no actual "protons" may move. It was, I believe, Benjamin Franklin who decided to declare that current move in a particular direction and, from the point of view of some physical matter actually moving, he got it wrong! Of course, he did not know anything about "protons" or "electrons". He was working with static electricity generators and, seeing a spark jump between his finger and the generator, assumed something was coming from the generator to his finger, not realizing that the "electrons" were actually going the other way. On a larger scale, he assumed that lightning was something going from the clouds to the earth. In fact, it is more often electrons going from the Earth to the cloud.