How exactly does electrolysis occur if the molar conductivities of cation and anion are different?
You are completely right that ions migrate at different rates, and the relative rates can be quantified by a property of the electrolyte called the transport number, which is the fraction of the current that can be accounted by one of the ions:
$$t_\circ^+ = \frac{\Lambda_\mathrm{m}^\circ(\ce{A+})}{\Lambda_\mathrm{m}^\circ(\ce{A+})+\Lambda_\mathrm{m}^\circ(\ce{B-})}$$
$$t_\circ^- = \frac{\Lambda_\mathrm{m}^\circ(\ce{B-})}{\Lambda_\mathrm{m}^\circ(\ce{A+})+\Lambda_\mathrm{m}^\circ(\ce{B-})}$$
(here $t_\circ^i$ designates a limiting transport number ie one in the limit of dilute electrolyte solution).
In the early days of the study of electrolytes, Hittorf developed a way of determining the ratio of transport numbers by measuring the concentrations of electrolytes in different parts of an electrolytic cell. Say the cell contains a cathode, an anode, and a bridging chamber. Cations are attracted toward the cathode chamber and depleted at the cathode, and anions are similarly depleted at the anode but also migrate into the anode chamber. The change in the concentration of cations in the cathode chamber is therefore proportional to the current at the electrode minus the fraction of current carried by cations replenishing the cation chamber:
$$\begin{align} \frac{dc_+}{dt} &amp;= \frac{I_+}{z_+F} - \frac{I}{z_+F} \\  &amp;= \frac{(t_+-1)I}{z_+F} \\  &amp;= \frac{-t_-I}{z_+F} \end{align}$$
A similar equation can be written for the anions. The resulting concentrations in the chambers can be related to the migration rates $v_i$ of the ions , which can in turn be related to the transport numbers as
$$\frac{t^+}{t^-} = \frac{|Z^+v^+|}{|Z^+v^-|}$$
Therefore from measurement of electrolyte concentrations in different parts of a cell it is possible to deduce the relative molar conductivities of the components of an electrolyte.
Because the counterions have different mobilities a concentration gradient will develop and this leads to a liquid junction potential. This results in a voltage drop within the cell that counters the EMF, such that the measured cell potential is not the ideal EMF. In Hittorf's experiment this is not a problem. Where it is a problem it might be avoided by using a salt bridge.
For a textbook that provides a reasonably clear introduction to electrochemistry see Atkins Physical Chemistry, from which I derived some of this answer.
Transport numbers are important in practical applications. See for instance Ref. 1 for an interesting thesis that focuses on the determination of transport numbers in cements.
References

J.A. Muka (1992). Transport Number Determination by Analysis of Transport of Aqueous Electrolytes Through Cementitious Materials Using Emf Method. University of Nairobi.