logical deduction - JonMark Perry's Grid Logic Puzzle

Consider the following puzzle type proposed by JonMark Perry.

Start with a square grid of arrows, each one pointing in one of the four cardinal directions. For example:

You start at the top left. You travel 1 step in the direction of the arrow you are on. Continue until you reach the bottom right.

Only there's a 'twist' - when you leave an arrow, it rotates 90° clockwise.

^{edited from JMP's original "1 or 2 steps" for simplicity of the problem}

Let's try to analyse this puzzle!

Is it always solvable, regardless of the initial setup of arrows in the grid? If not, are there conditions for when it is solvable? Does the answer vary according to the size of the grid?

_{Disclaimer: I don't know the answer to this question, and nor, it seems, does JMP himself.}