  In this paper, we study the Cauchy problem for a generalized cross-coupled
Camassa-Holm system with peakons and higher-order nonlinearities. By the
transport equation theory and the classical Friedrichs regularization method,
we obtain the local well-posedness of solutions for the system in
nonhomogeneous Besov spaces $B^s_{p,r}\times B^s_{p,r}$ with $1\leq p,r \leq
+\infty$ and $s>\max\{2+\frac{1}{p},\frac{5}{2}\}$. Moreover, we construct the
local well-posedness in the critical Besov space $B^{5/2}_{2,1}\times
B^{5/2}_{2,1}$ and the blow-up criteria. In the paper, we also consider the
well-posedness problem in the sense of Hadamard, non-uniform dependence, and
H\"older continuity of the data-to-solution map for the system on both the
periodic and the non-periodic case. In light of a Galerkin-type approximation
scheme, the system is shown well-posed in the Sobolev spaces $H^s\times
H^s,s>5/2$ in the sense of Hadamard, that is, the data-to-solution map is
continuous. However, the solution map is not uniformly continuous. Furthermore,
we prove the H\"older continuity in the $H^r\times H^r$ topology when $0\leq r<
s$ with H\"older exponent $\alpha$ depending on both $s$ and $r$.
