In QFT, when computing cross sections one calculates the $T$ matrix elements (probability amplitudes that the interaction Hamiltonian takes us from the initial state $|\mathbf{k}_A,\mathbf{k}_B\big>$ to the final state $|\mathbf{p}_1,\mathbf{p}_2...\big>$):
$$\big<\mathbf{p}_1\mathbf{p}_2...|iT|\mathbf{k}_A\mathbf{k}_B\big>=(2\pi)^4\delta^4(k_A+k_B-\sum p_f)i\mathcal{M}(k_A,k_B\rightarrow p_f)$$
(This is discused in Peskin Eq. 4.73.) In free space 4-momentum is conserved, and the quantity $\mathcal{M}$ appears in the expression of the differential cross section as: $$d\sigma=\frac{1}{2E_AE_B|v_A-v_B|}|\mathcal{M}(p_A,p_B\rightarrow \{p_f\})|^2(2\pi)^4\delta^4(k_A+k_B-\sum p_f),$$ where $p_A$ and $p_B$ are the values where the initial momenta distributions are concentrated. $E_A$ and $E_B$ ($v_A$ and $v_B$) are the energies (velocities) of the intial particles.
What happens when momentum in a given direction (say, the $z$ axis) is not conserved? I imagine we would get something like $$\big<\mathbf{p}_1\mathbf{p}_2...|iT|\mathbf{k}_A\mathbf{k}_B\big>=(2\pi)^3\delta(E_A+E_B-\sum E_f)\delta^2(\mathbf{p}_{\perp,A}+\mathbf{p}_{\perp,B}-\sum \mathbf{p}_{\perp,f})i\mathcal{M},$$ where the subscript $\perp$ indicates the components of momenta perpendicular to the $z$-axis. How can we define cross sections in that case?