Let (*Y*, *d*) be a nontrivial metric space and (*Y*, *g*_{1,∞}) be a nonautonomous discrete dynamical system given by sequences $(g_{l})_{l = 1}^{\infty }$ of continuous maps *g*_{l} : *Y* → *Y* and let $\mathcal {F}$, $\mathcal {F}_{1}$ and $\mathcal {F}_{2}$ be given shift-invariant Furstenberg families. In this paper, we study stronger forms of transitivity and sensitivity for nonautonomous discrete dynamical systems by using Furstenberg family. In particular, we discuss the $\mathcal {F}$-transitivity, $\mathcal {F}$-mixing, $\mathcal {F}$-sensitivity, $\mathcal {F}$-collective sensitivity, $\mathcal {F}$-synchronous sensitivity, $(\mathcal {F}_{1},\mathcal {F}_{2})$-sensitivity and $\mathcal {F}$-multi-sensitivity for the system (*Y*, *g*_{1,∞}) and show that under the conditions that *g*_{j} is semi-open and satisfies *g*_{j} ∘ *g* = *g* ∘ *g*_{j} for each *j* ∈ {1, 2, ⋯ } and that
$$\sum\limits_{j = 1}^{\infty}D(g_{j},g) $$exists (i.e., $\sum \limits _{j = 1}^{\infty }D(g_{j},g)<+\infty $), the following hold:
(1)

(*Y*, *g*_{1,∞}) is $\mathcal {F}$-transitive if and only if so is (*Y*, *g*).

(2)

(*Y*, *g*_{1,∞}) is $\mathcal {F}$-mixing if and only if so is (*Y*, *g*).

(3)

(*Y*, *g*_{1,∞}) is $\mathcal {F}$-sensitive if and only if so is (*Y*, *g*).

(4)

(*Y*, *g*_{1,∞}) is $(\mathcal {F}_{1},\mathcal {F}_{2})$-sensitive if and only if so is (*Y*, *g*).

(5)

(*Y*, *g*_{1,∞}) is $\mathcal {F}$-collectively sensitive if and only if so is (*Y*, *g*).

(6)

(*Y*, *g*_{1,∞}) is $\mathcal {F}$-synchronous sensitive if and only if so is (*Y*, *g*).

(7)

(*Y*, *g*_{1,∞}) is $\mathcal {F}$-multi-sensitive if and only if so is (*Y*, *g*).

The above results extend the existing ones.