Let $X$ be a curve over a scheme $k$: Let $H_{n,X}$ be the punctual scheme of $X$ parametrizing finite subschemes of degree $n$, and le $\varphi_{n,X}: H_{n,X} \rightarrow X^{(n)}$ be the Hilbert-Chow morphism. For a closed point $x \in X$, be the local punctual Hilbert scheme $H_{n,X,x} = \varphi_{n,X}^{-1}(nx)$ is known in many cases (for example, curves with nodal singularities).

Suppose that $x_1, \dots, x_r$ are distinct points of $X$. Is it possible to deduce something on the geometry of $\varphi_{n,X}^{-1}\left( \sum n_i x_i \right)$ from the geometry of the $H_{n,X,x_i}$'s?

For example, on a node $x$, any cluster $Z \in H_{n,X,x}$ which is not defined by a single local equation is a limit of subschemes defined by single equations (Example 2.1 in Bertin's The punctual Hilbert scheme: an Introduction). If $x_1, \dots, x_r$ are nodes, does a similar fact hold also for $\varphi_{n,X}^{-1}\left( \sum n_i x_i \right)$?