I'll answer only for SDS-Page, which is the system I am most familiar with.
With a discontinuous buffer system, such as the well-known Laemmli system, resistance increases during electrophoresis, as (very mobile) chloride ions are replaced by glycinate (glycine ions).
From Ohms law:
Voltage (V) = Current (I) x Resistance (R)
and the definition of power (Watts):
Watts (W) = Current (I) x Voltage (V) = I2 x R
- At constant current electrophoresis will proceed at a uniform rate, but as the resistance increases so too will the voltage, and so too will the amount of heat generated.
In some discontinuous buffer systems, such as the Tricine/SDS system due to Schägger & von Jagow, this may be enough to crack the gel plates!
- At constant voltage, the current will decrease during electrophoresis, and so too will the amount of head generated, but (as a consequence of the decrease in current) so too will the rate of migration of the sample.
Laemmli, U.K. (1970) Cleavage of structural proteins during the assembly of the head of bacteriophage T4. Nature 227, 680-685. [PubMed]
Schägger, H. & von Jagow, G (1987).
Tricine-sodium dodecyl sulfate-polyacrylamide gel electrophoresis for the separation of proteins in the range from 1 to 100 kDa. Anal Biochem 166, 368-379.[PubMed]