I have phrased similarly another question about how physicists knew that two charges exist, positive and negative. The purpose of the question is not necessarily to educate me historically. It's just that I wish to know about classical subjects without making the atomic assumption.
I know that electrons (elementary negatively charged particles) move in contrast with protons (elementary positively charged particles) because electrons have small mass and orbit the nucleus while protons are stuck in the nucleus of atoms. Roughly at least!
Charging by induction works because of the transfer of electrons, negative charge between a conductive sphere and the ground. But it could be very well explained by the transfer of either positive or negative charge. But only the later happens.
Was there any (thought) experiment to show that only negative charges happen to move/transfer?
Answer
The Hall Effect shows that negative charge is moving. 
In the Hall effect, one passes a current through a wide strip of metal exposed to a perpendicular magnetic field. If positive charges moved, we'd expect the positive charges to be travelling in the same direction as $\vec{I}$, and the magnetic force $q\vec{v}\times\vec{B}$ would be to the right. Thus, we'd expect an accumulation of positive charges on the right of the strip.
However, if negative charges move, we expect they move in the opposite direction of $\vec{I}$, and the magnetic force $q\vec{v}\times\vec{B}$ would again be to the right. This time, we expect an accumulation of negative charges on the right side of the strip.
If you actually do the experiment, you find that the right side of the strip is negatively charged, as shown in the picture. So negative charges are doing the moving!
Image Source: https://www.nde-ed.org/EducationResources/CommunityCollege/MagParticle/Physics/Measuring.htm
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