We study (Euclidean) k-median and k-means with constraints in the streaming model. There have been recent efforts to design unified algorithms to solve constrained k-means problems without using knowledge of the specific constraint at hand aside from mild assumptions like the polynomial computability of feasibility under the constraint (compute if a clustering satisfies the constraint) or the presence of an efficient assignment oracle (given a set of centers, produce an optimal assignment of points to the centers which satisfies the constraint). These algorithms have a running time exponential in k, but can be applied to a wide range of constraints. We demonstrate that a recently proposed technique for solving a specific constrained k-means problem, namely fair k-means clustering, actually implies streaming algorithms for all these constraints. The technique uses the computation of so-called coresets. A coreset is a small summary that approximately preserves the cost of all feasible solutions. A special type of coreset for which constructions were proposed at the beginning of the coreset era actually satisfy a very strong type of guarantee which implies their applicability to general constrained k-means problems. The drawback is that these constructions introduce an exponential dependence on the dimension d. However, for constant d, one immediately gets a streaming algorithm which computes a coreset for any constrained k-means problem (under above mild assumptions) and which can be combined with approximation algorithms to get a streaming approximation algorithm. We note that our paper builds heavily on previous work, yet we believe that it is worthwhile to know that streaming coresets for constrained k-means (and Euclidean k-median) clustering can be obtained fairly easily independently of the specific constraint.