Equivariant quantum channels are completely positive trace-preserving maps intertwining representations of a group G. Lieb and Solovej (2014) studied traces of the functional calculus of equivariant quantum channels for SU(2) to establish a Wehrl-type inequality for integrals of convex functions of matrix coefficients. In particular, they showed that coherent highest weight states minimize the Wehrl entropy, solving a long-standing conjecture. These quantum channels are defined by projecting onto the leading component in the decomposition of the tensor product of two irreducible representations of SU(2). It is proved by Lieb and Solovej that the aforementioned integral of a convex function is the limit of the trace of the functional calculus of these equivariant quantum channels. I introduce new equivariant quantum channels for all the components in the tensor product and generalize their limit formula. I do this by realizing representations of SU(2) as reproducing kernel spaces and using explicit projections onto components of the tensor product. This allows me to pass from the trace to an integral to derive a limit formula and prove that the limit can be expressed using Berezin transforms, which are closely related to quantization on Kähler manifolds.