An important result by Dudley from the 1960s states that every algebraic homomorphism from a locally compact Hausdorff group to a free group is continuous. A generalization by Morris and Nickolas shows that this remains true for arbitrary free products of groups unless the image of the homomorphism is small. We are able to show that this pattern remains true for many groups studied in geometric group theory if their torsion subgroups are small. In particular, we show that every surjective algebraic homomorphism from a locally compact Hausdorff group into the automorphism group of a right-angled Artin group $Aut(A_\Gamma)$ is continuous.