  For a topological dynamical system $(X, T)$, $l\in\mathbb{N}$ and $x\in X$,
let $N_l(X)$ and $L_x^l(X)$ be the orbit closures of the diagonal point
$(x,x,\ldots,x)$ ($l $ times) under the actions $\mathcal{G}_{l}$ and $\tau_l $
respectively, where $\mathcal{G}_{l}$ is generated by $T\times T\times \ldots
\times T$ ($l $ times) and $\tau_l=T\times T^2\times \ldots \times T^l$. In
this paper, we show that for a minimal system $(X,T)$ and $l\in \mathbb{N}$,
the maximal $d$-step pro-nilfactor of $(N_l(X),\mathcal{G}_{l})$ is
$(N_l(X_d),\mathcal{G}_{l})$, where $\pi_d:X\to X/\mathbf{RP}^{[d]}=X_d,d\in
\mathbb{N}$ is the factor map and $\mathbf{RP}^{[d]}$ is the regionally
proximal relation of order $d$.
  Meanwhile, when $(X,T)$ is a minimal nilsystem, we also calculate the
pro-nilfactors of $(L_x^l(X),\tau_l)$ for almost every $x$ w.r.t. the Haar
measure. In particular, there exists a minimal $2$-step nilsystem $(Y,T)$ and a
countable set $\Omega\subset Y$ such that for $y\in Y\backslash \Omega$ the
maximal equicontinuous factor of $(L_y^2(Y),\tau_2)$ is not
$(L_{\pi_1(y)}^2(Y_{1}),\tau_2)$.
