Right moving turing machines and FSA's
As Yuval indicated in the comment, a right-moving TM is already essentially a FA. Here's the construction:

Given a TM move rule $\delta(q, x)=(p, y, d)$ where, as usual, $q$ is the current state, $x$ is the contents of the current cell, $p$ is the new state, $y$ is the symbol replacing $x$ in the current cell, and $d$ is the direction to move the head. Since the machine can only move right, it is immaterial what gets overwritten on the current cell and since $d$ is always move right, we can ignore both of these, leaving us with moves of the form $\delta(q, x)=(p)$, i.e., the moves for a FA.

The construction above is for a DFA, the one for a NFA is essentially the same.