Let K be an arbitrary field and Cn a relatively free algebra of rank n. In particular, as Cn we may treat a polynomial algebra Pn, a free associative algebra An, or an absolutely free algebra Fn. For the algebras Cn = Pn, An, Fn, it is proved that every finitely generated subgroup G of a group TCn of triangular automorphisms admits a faithful matrix representation over a field K; hence it is residually finite by Mal’tsev’s theorem. For any algebra Cn, the triangular automorphism group TCn is locally soluble, while the unitriangular automorphism group UCn is locally nilpotent. Consequently, UCn is local (linear and residually finite). Also it is stated that the width of the commutator subgroup of a finitely generated subgroup G of UCn can be arbitrarily large with increasing n or transcendence degree of a field K over the prime subfield.