In simple terms, you can think of the reaction as a straight substitution of OH- for Cl-:
$$ ACl_3+3H_2O\to A(OH)_3+3HCl$$
This works as advertised for arsenic and phosphorus, although phosphorus mainly converts to the tautomeric (OH)2HP=O, phosphorous acid. The problem with the nitrogen case is that N(OH)3 is completely unstable. You're never going to see it in solution. Instead, you might expect to see a complex mixture of NH2OH or various nitrogen oxides, as well as a mixture of chloramines. Nitrogen in its +3 oxidation state is also a pretty good oxidizer,