  For a continuous self-map $f$ on a compact interval $I$ and the induced map
$\hat f$ on the space $\mathcal{M}(I)$ of probability measures, we obtain a
sharp condition to guarantee that $(I,f)$ is transitive if and only if
$(\mathcal{M}(I),\hat f)$ is transitive. We also show that the sensitivity of
$(I,f)$ is equivalent to that of $(\mathcal{M}(I),\hat f)$. We prove that
$(\mathcal{M}(I),\hat f)$ must have infinite topological entropy for any
transitive system $(I,f)$, while there exists a transitive non-autonomous
system $(I,f_{0,\infty})$ such that $(\mathcal{M}(I),\hat f_{0,\infty})$ has
zero topological entropy, where $f_{0,\infty}=\{f_n\}_{n=0}^\infty$ is a
sequence of continuous self-maps on $I$. For a continuous self-map $f$ on a
general compact metric space $X$, we show that chain transitivity of $(X, f)$
implies chain mixing of $(\mathcal{M}(X),\hat f)$, and we provide two
counterexamples to demonstrate that the converse is not true. We confirm that
shadowing of $(X,f)$ is not inherited by $(\mathcal{M}(X),\hat f)$ in general.
For a non-autonomous system $(X,f_{0,\infty})$, we prove that if
$(\mathcal{M}(X),\hat{f}_{0,\infty})$ is weak mixing of order $n$, then so is
$(X,f_{0,\infty})$ for any $n\geq2$; while there exists $(X,f_{0,\infty})$ such
that it is weak mixing of order $2$ but $(\mathcal{M} (X),\hat{f}_{0,\infty})$
is not. We then prove that Li-Yorke chaos (resp., distributional chaos) of
$(X,f_{0,\infty})$ carries over to $(\mathcal{M}(X),\hat f_{0,\infty})$, and
give an example to show that $(X,f)$ and $(\mathcal{M}(X),\hat f)$ may have no
Li-Yorke pair simultaneously. We also prove that if $f_n$ is surjective for all
$n\geq 0$, then chain mixing of $(\mathcal{M}(X),\hat f_{0,\infty})$ always
holds true, and shadowing of $(\mathcal{M}(X),\hat f_{0,\infty})$ implies
mixing of $(X, f_{0,\infty})$.
