Given $f_1,...,f_n \in \mathbb{F}_q[x_1,...,x_m]$ such that $deg(f_i) \leq d < q$. Suppose we have for some $1 \leq j \leq m$

$$ trdeg_{\mathbb{F}(x_1,...,x_j)}\{f_1...,f_n\} = r$$

Let $z_1,...,z_j$ be chosen uniformly and independently at random from $\mathbb{F}_q$ then what is

$$ Pr_{z_1,...,z_j}[trdeg_{\mathbb{F}}\{ f_1(z_1,..,z_j,x_{j+1},..,x_m),...,f_n(z_1,...,z_j,x_{j+1},..,x_m) \} = r ]?$$

Ideally is the probability large, i.e. not inverse-polynomial in n,m or worse? This probability would be very helpful in my research and any help is appreciated.