We consider products of $ψ$ classes and products of $ω$ classes on $\overline{M}_{0,n+3}$. For each product, we construct a flat family of subschemes of $\overline{M}_{0,n+3}$ whose general fiber is a complete intersection representing the product, and whose special fiber is a generically reduced union of boundary strata. Our construction is built up inductively as a sequence of one-parameter degenerations, using an explicit parametrized collection of hyperplane sections. Combinatorially, our construction expresses each product as a positive, multiplicity-free sum of classes of boundary strata. These are given by a combinatorial algorithm on trees we call 'slide labeling'. As a corollary, we obtain a combinatorial formula for the $κ$ classes in terms of boundary strata. For degree-$n$ products of $ω$ classes, the special fiber is a finite reduced union of (boundary) points, and its cardinality is one of the multidegrees of the corresponding embedding $Ω_n: \overline{M}_{0,n+3}\to \mathbb{P}^1\times \cdots \times \mathbb{P}^n$. In the case of the product $ω_1\cdots ω_n$, these points exhibit a connection to permutation pattern avoidance. Finally, we show that in certain cases, a prior interpretation of the multidegrees via tournaments can also be obtained by degenerations.