  We prove a local Douglas formula for higher order weighted Dirichlet-type
integrals. With the help of this formula, we study the multiplier algebra of
the associated higher order weighted Dirichlet-type spaces $\mathcal
H_{\pmb\mu},$ induced by an $m$-tuple $\pmb \mu =(\mu_1,\ldots,\mu_{m})$ of
finite non-negative Borel measures on the unit circle. In particular, it is
shown that any weighted Dirichlet-type space of order $m,$ for $m\geqslant 3,$
forms an algebra under pointwise product. We also prove that every non-zero
closed $M_z$-invariant subspace of $\mathcal H_{\pmb\mu},$ has codimension $1$
property if $m\geqslant 3$ or $\mu_2$ is finitely supported. As another
application of local Douglas formula obtained in this article, it is shown that
for any $m\geqslant 2,$ weighted Dirichlet-type space of order $m$ does not
coincide with any de Branges-Rovnyak space $\mathcal H(b)$ with equivalence of
norms.
