We consider a second order differential operator $\mathscr{A}$ on an (typically unbounded) open and Dirichlet regular set $Ω\subset \mathbb{R}^d$ and subject to nonlocal Dirichlet boundary conditions of the form \[ u(z) = \int_Ωu(x)μ(z, dx) \quad \mbox{ for } z\in \partial Ω. \] Here, $μ: \partialΩ\to \mathscr{M}(Ω)$ is a $σ(\mathscr{M}(Ω), C_b(Ω))$-continuous map taking values in the probability measures on $Ω$. Under suitable assumptions on the coefficients in $\mathscr{A}$, which may be unbounded, we prove that a realization $A_μ$ of $\mathscr{A}$ subject to the nonlocal boundary condition, generates a (not strongly continuous) semigroup on $L^\infty(Ω)$. We also establish a sufficient condition for this semigroup to be Markovian and prove that in this case, it enjoys the strong Feller property. We also study the asymptotic behavior of the semigroup.