The mole fractions of the components of vapours aren't the same as those of the liquid-liquid solution (unless the solution is azeotropic), this implies a change in the composition of the liquid solution when vapours arise. Consequently, the partial vapour pressures of the components would also change and there would be a messed up cycle. There has to be something that I'm understanding wrongly and I'd like to be told what it is.
It is not a messed up cycle. Let's consider what happens with an example.
If you heat up a mixture containing $10$% ethanol + $90$% water, the mixture boils at $92$°C, producing a vapor containing $50$% ethanol + $50$% water. With this operation the liquid loses more ethanol than water. So its concentration in ethanol decreases, and it is necessary to heat it up to a higher temperature to maintain the process of evaporation. When about one tenth of the liquid has distilled, the residual liquid contains still $5$% ethanol, and it is necessary to heat it up to $95$°C to obtain and continue boiling. At $95$°C, the vapor emitted contains $35$% ethanol. At $98°$C, the liquid contains still $2$% ethanol and the vapor $20$% ethanol. When the last drops are vaporized, the temperature is $100$°C and the last drops are made of pure water.