  In this paper we discuss $R$-matrix-valued Lax pairs for ${\rm sl}_N$
Calogero-Moser model and their relation to integrable quantum long-range spin
chains of the Haldane-Shastry-Inozemtsev type. First, we construct the
$R$-matrix-valued Lax pairs for the third flow of the classical Calogero-Moser
model. Then we notice that the scalar parts (in the auxiliary space) of the
$M$-matrices corresponding to the second and third flows have form of special
spin exchange operators. The freezing trick restricts them to quantum
Hamiltonians of long-range spin chains. We show that for a special choice of
the $R$-matrix these Hamiltonians reproduce those for the Inozemtsev chain. In
the general case related to the Baxter's elliptic $R$-matrix we obtain a
natural anisotropic extension of the Inozemtsev chain. Commutativity of the
Hamiltonians is verified numerically. Trigonometric limits lead to the
Haldane-Shastry chains and their anisotropic generalizations.
