Marginalized non Markov Chapman kolmogorov equation

Question

The usual Chapman kolmogorov equation states $$\int dx_1 p(x_2|x_1)p(x_1|x_0)=p(x_2|x_0)$$ Which means we can also identify $$\int dx_0 p(x_2|x_1)p(x_1|x_0)p(x_0)=p(x_2|x_1)p(x_1)$$ Now, because $$\int dx_0 p(x_2,x_1,x_0)=p(x_2,x_1)$$ We can write the non Markov case as $$\int dx_0 p(x_3|x_2,x_1,x_0)p(x_2|x_1,x_0)p(x_1|x_0)p(x_0)=p(x_3|x_2,x_1)p(x_2|x_1)p(x_1)$$ The problem I have is that I have an integral in the form $$\int dx_0 p(x_3|x_2,x_1,x_0)p(x_2|x_1)p(x_1|x_0)p(x_0)$$ Where $$p(x_2|x_1)=\frac{\int dx_0 p(x_2,x_1,x_0)}{\int dx_0 p(x_1,x_0)}$$ My question is can I simply marginalize over the nuisance variable as before and get $$\int dx_0 p(x_3|x_2,x_1,x_0)p(x_2|x_1)p(x_1|x_0)p(x_0)=p(x_3|x_2,x_1)p(x_2|x_1)p(x_1)$$ And if not why not and can we write it as anything else?