Does the thing at one end of the wire get heavier and the thing at the other end lighter?
Because of the mass of the electrons moved from one thing to the other.
Because of the mass of the electrons moved from one thing to the other.
Imagine a simple circuit, say a light bulb and a battery. Electrons move from the negative terminal, through the bulb, and back to the battery. The net change in number of electrons at any one point is zero. The energy isn't in the electrons themselves, but in the motion of those electrons. Electrons in must equal erlctrons out.
Even a battery doesn't store electrons. It uses the energy carried by those electrons to reverse a chemical reaction. The energy is stored chemically.
If you think about it, the electrons belong to the physical materials in the circuit. You can't really add or remove electrons* as electron count is a fundamental property of those atoms. If you somehow removed electrons from the system, you'd be changing those atoms and the system would no longer be able to pass current at all.
*you can, of course ionize atoms by adding or removing electrons, but that's not exactly what happens in electric circuits
Electrons are not electricity, they just carry it. Kind of. It's really complicated.
Positive charge carriers do not actually exist[0]. There's only electrons and holes they can go into. We can talk about the movement of holes, but that's a virtual charge carrier at best.
Conventional current is just a convention. It's what we started with (because Franklin was wrong) and it's too much effort to change now. In practice, the distinction almost never matters. Sometimes it does, but not enough that it's worth overhauling the entire field of electronics.
[0] of course positrons and protons exist, but they aren't relevant to electronic circuits
However, if you're talking about static electricty, you can actually create a mass imbalance by taking a few electrons away from one side or putting some electrons on the other. It's a very, very, very small change in mass.
If the answer is yes then we have moving electrons.
If the answer is no then maybe we don't.
Imagine a different scenario, where the pipe ends in a big box: This time the box does fill up with water, and gets heavier.
Mapping the analogy from water back to electrons: a loop of pipe is like a loop of wire and a battery; while the pipe ending with a box becomes a capacitor or antenna, and that will leak[0] before you can measure the mass change — but technically yes the the mass of any given wall of a capacitor or of an antenna will be very slightly changed by this sort of thing.
For a sense of scale, to get a total charge of 1 coulomb using electrons, the mass of those electrons will be about 5.7 nanograms, and trying to squeeze that much charge into the last millimetre of some length of a wire 1mm in cross section diameter, involves about 60% of the energy in this explosion: https://www.youtube.com/watch?v=wqKn_3iJOP4
As nothing gets close to being able to hold that kind of energy, even if you're trying to accumulate a lot of excess electrons, those electrons leak well before even coming close to nanograms of excess mass.
[0] The bit on the top with the sparks coming out is a capacitive electrode: https://en.wikipedia.org/wiki/Tesla_coil
You can move whole charged atoms, that's a form of electricity too, and it can add observable amount of mass, like with electroplating or welding. But these very quickly turn electrically neutral after depositing.