  In this paper, we study the Cauchy problem for a two-component higher order
Camassa-Holm systems with fractional inertia operator
$A=(1-\partial_x^2)^r,r\geq1$, which was proposed by Escher and Lyons. By the
transport equation theory and Littlewood-Paley decomposition, we obtain that
the local well-posedness of solutions for the system in nonhomogeneous Besov
spaces $B^s_{p,q}\times B^{s-2r+1}_{p,q}$ with $1\leq p,q \leq +\infty$ and the
Besov index $s>\max\left\{2r +\frac{1}{p},2r+1-\frac{1}{p}\right\}$. Moreover,
we construct the local well-posedness in the critical Besov space
$B^{2r+\frac{1}{2}}_{2,1}\times B^{\frac{3}{2}}_{2,1}$. On the other hand, the
propagation behaviour of compactly supported solutions is examined, namely
whether solutions which are initially compactly supported will retain this
property throughout their time of evolution. Moreover, we also establish the
persistence properties of the solutions to the two-component Camassa-Holm
equation with $r=1$ in weighted $L_{\phi}^p:=L^p(\mathbb{R},\phi^p(x)dx)$
spaces for a large class of moderate weights.
