How does the composition of volcanic rock change as it makes its ascent towards the surface?
Does the composition of the volcanic rock change drastically according to the distance between the magma column and the mouth of the volcano at the time the rock is formed? 


The metal composition of the magma will not change drastically. By metal I mean the basic constituents of the rock: mostly silicon but also iron, calcium, aluminum, magnesium, etc. I will add that there are processes that can affect metal composition (chiefly mineral fractionation) but I think its influence is minor when you're talking the the uppermost part of the volcano where it erupts.

What can and does change is the volatile composition. The magma is depleted of gases. Take for example a bottle of coke. When you open it, the CO2 bubbles out. The coke itself remains coke, but its chemistry changes a bit (and it tastes bad) because the CO2 is out. The magma may also change oxidation state. This, together with the volatile degassing may eventually affect the minerals that crystallise from the magma.


  (Or) do the rocks melt and recrystallize frequently as the boundary moves, only attaining their final composition before they are ejected into the air? 


No. Once a rock crystallises you are going to need quite a lot of energy to melt it again, and this is not a likely process to occur in a system open to the atmosphere where heat is free to leave the volcanic system.