(As Will Sawin has answered the question, in a comment above, I record here the detailed answer. Still, I'd like to know some references where all this is written.)
1. As some of the multi-degrees can coincide, arrange them as follows: $d_1^{(1)}=d_1^{(2)}=\cdots=d_1^{(r_1)}< d_2^{1}=\cdots=d_2^{(r_2)}<\cdots<d_k^{(r_k)}$. Let $GCI_{d^{r_1}_1\cdots d^{r_k}_k}$ denote the parameter space of globally complete intersections of the multidegree as above. This space is the chain of fibrations as follows.
$GCI_{d^{(r_1)}_1}=Gr(\mathbb{P}^{r_1-1},|\mathcal{O}_{\mathbb{P}^n}(d_1)|)$. And the forgetful projection $GCI_{d^{(r_1)}_1\cdots d^{(r_{j+1})}_{j+1}}\to GCI_{d^{(r_1)}_1\cdots d^{(r_{j})}_{j}}$ is the projectivization of the vector bundle. Its fibre over a point $I=(f_{1,d_1},\dots,f_{r_1,d_1},f_{1,d_2},\dots,f_{r_2,d_2},\dots,f_{r_j,d_j})\subset k[x_0,\dots,x_n]$ of $GCI_{d^{(r_1)}_1\cdots d^{(r_{j})}_{j}}$ is: $Gr(\mathbb{P}^{r_{j+1}-1},\mathbb{P}(H^0(\mathcal{O}_{\mathbb{P}^n}(d_{j+1}))/I[d_{j+1}])$ . Here $I[d_{j+1}]$ is the homogeneous part of $I$ consisting of polynomials of degree $d_{j+1}$.
This is a smooth and not too complicated parameter space.
2. Many properties of the discriminant, $\Delta\subset GCI_{d^{r_1}_1\cdots d^{r_k}_k}$ can be obtained from the local consideration (Looijenga's book). It is a reduced divisor. To prove the irreducibility one considers the standard Nash modification, i.e. the incidence variety, $\tilde{\Delta}\subset \mathbb{P}^n\times GCI_{d^{r_1}_1\cdots d^{r_k}_k}$, consisting of pairs: the complete intersection and (one of) its singular point. The projection $\tilde{\Delta}\to\Delta$ is birational, while all the fibres of the projection onto $\mathbb{P}^n$ are linear spaces. Thus $\tilde{\Delta}$ is smooth, in particular $\Delta$ is irreducible.