The physics
Current is the charge that crosses a section of wire each second. Each electron carries a charge e = 1.6 × 10⁻¹⁹ C.
I = Q / t = N e / t I = dQ / dt
Free electrons in a metal rush about at random. A field adds a slow, steady drift on top. That drift carries the current.
I = n e A v J = I / A = n e v
A wire with no branches carries the same current through every section. Where the wire is thin, the electrons drift faster.
On a cell, the field in the wire is E = V / L. The drift speed is v = μE. Here μ is the mobility of the metal.
v = μ E = E / (ρ n e)
A warmer metal has a bigger ρ. So its electrons drift slower on the same cell.