Given a family of divisors $D_t$ on varieties $X_t$, there are examples that show that bigness is not well behaved (e.g. example 2.2.13 in Positivity 1, shows we can have a special fiber where $D_0$ is big, while for general $t$ $D_t$ has negative Kodaira dimension).

Can we argue any openness if we know a bit more? Say that the base of the family is $U$, and for a countable set $\lbrace t_i \rbrace$ such that $\overline{\lbrace t_i \rbrace}=U$ we know that $D_{t_i}$ is big. Does then the general $D_t$ have to be big?