This paper deals with the Cauchy problem for the interacting system of the Camassa–Holm and Degasperis–Procesi equations $$\begin{aligned} m_t=-3m(2u_x+v_x)-m_x(2u+v), n_t=-2n(2u_x+v_x)-n_x(2u+v), \end{aligned}$$ where $$m=u-u_{xx}$$ and $$n= v-v_{xx}$$ . By the transport equations theory and the classical Friedrichs regularization method, the local well-posedness of solutions for this system in nonhomogeneous Besov spaces $$B^s_{p,r}\times B^s_{p,r}$$ with $$1\le p,r \le +\infty $$ and $$s>\max \left\{ 2+\frac{1}{p},\frac{5}{2}\right\} $$ is obtained, and the local well-posedness in critical Besov space $$B^{5/2}_{2,1}\times B^{5/2}_{2,1}$$ is also established. Moreover, by the approach for approximate solutions and well-posedness estimates, we obtain two sequences of solution for this equation, which are bounded in the Sobolev space $$H^s({\mathbb {R}})\times H^s({\mathbb {R}})$$ with $$s>5/2$$ , and the distance between the two sequences is lower-bounded by a positive constant for any time t, but converges to zero at the initial time. This implies that the solution map is not uniformly continuous.