For this answer the assumption that the points $x_i^n$ are distinct is not necessary. The restriction to $[0,1]$ is also not necessary. The answer applies to Radon probability measures on any complete metric space.

On the set of Borel probability measures on $[0,1]$, the topology in question is given by the Kantorovich-Rubinshtein metric $d_0$. See V.I. Bogachev, "Measure theory", Chapter 8. In this metric the set of probability measures is complete. So a sequence converges if and only if it is Cauchy.

The Kantorovich-Rubinshtein metric is the optimal transport metric. The distance $d_0(\rho,\sigma)$ is the least cost of moving some divisible matter from a state distributed according to $\rho$ to one distributed according to $\sigma$, when the cost of moving a unit amount from $x\in[0,1]$ to $y\in[0,1]$ is the distance $|x-y|$.

So for discrete measures one can describe the **necessary and sufficient** Cauchy condition in some more or less combinatorial ways. The following formulation describes a "transport plan" from $\rho_m$ to $\rho_n$; we split $\rho_m$ and $\rho_n$ into smaller pieces which are then paired one-to-one, describing what is transported where: 

$(1) \;\; \forall \varepsilon>0 \;\exists M \;\forall m,n > M \;\exists K=K(m,n)
\;\exists y_1,y_2,...,y_K,z_1,z_2,...,z_K \in[0,1]$   
$\exists \alpha_1,\alpha_2,...,\alpha_K\in[0,1]
\;\;\rho_m = \sum_{i=1}^K \alpha_i \delta_{y_i} , 
\;\;\rho_n = \sum_{i=1}^K \alpha_i \delta_{z_i}, \;\;
\sum_{i=1}^K \alpha_i |y_i - z_i | < \varepsilon$.

From this one can derive various **sufficient** conditions, depending on what one desires. For example:

$(2) \;\; \forall \varepsilon>0 \;\exists M \;\forall n>m > M 
\;\exists y_1,y_2,...,y_{N_n} \in[0,1] \;\;
\exists \alpha_1,\alpha_2,...,\alpha_{N_n}\in[0,1]$   
$\rho_m = \sum_{i=1}^{N_n} \alpha_i \delta_{y_i}, 
\;\sum_{i=1}^{N_n} \alpha_i |x_i - y_i | < \varepsilon$.

A variation of condition (2) may be useful when the sets $X_n$ form the levels of a tree, so that each point in $X_{n+1}$ is linked to exactly one point in $X_n$. The transport plan then follows the edges of the tree.