  Asymptotic boundary KZB equations describe the consistency conditions of
degenerations of correlation functions for boundary Wess-Zumino-Witten-Novikov
conformal field theory on a cylinder. In the first part of the paper we define
asymptotic boundary KZB operators for connected real semisimple Lie groups G
with finite center. We prove their main properties algebraically using
coordinate versions of Harish-Chandra's radial component map. We show that
their commutativity is governed by a system of equations involving coupled
versions of classical dynamical Yang-Baxter equations and reflection equations.
We use the coordinate radial components maps to introduce a new class of
quantum superintegrable systems, called quantum Calogero-Moser spin chains. A
quantum Calogero-Moser spin chain is a mixture of a quantum spin Calogero-Moser
system associated to the restricted root system of G and an one-dimensional
spin chain with two-sided reflecting boundaries. The asymptotic boundary KZB
operators provide explicit expressions for its first order quantum
Hamiltonians. We also explicitly describe the Schr\"odinger operator.
