  The generation of $k$-designs (pseudorandom distributions that emulate the
Haar measure up to $k$ moments) with local quantum circuit ensembles is a
problem of fundamental importance in quantum information and physics. Despite
the extensive understanding of this problem for ordinary random circuits, the
crucial situations where symmetries or conservation laws are in play are known
to pose fundamental challenges and remain little understood. We construct, for
the first time, explicit local unitary ensembles that can achieve high-order
unitary $k$-designs under transversal continuous symmetry, in the particularly
important SU$(d)$ case. Specifically, we define the Convolutional Quantum
Alternating group (CQA) generated by 4-local SU$(d)$-symmetric Hamiltonians as
well as associated 4-local SU$(d)$-symmetric random unitary circuit ensembles,
and prove that they form and converge to SU$(d)$-symmetric $k$-designs,
respectively, for all $k < n(n-3)/2$ with $n$ being the number of qudits. A key
technique that we employ to obtain the results is the Okounkov--Vershik
approach to $S_n$ representation theory. To study the convergence time of the
CQA ensemble, we develop a numerical method using the Young orthogonal form and
$S_n$ branching rule. We provide strong evidence for a subconstant spectral gap
and certain convergence time scales of various important circuit architectures,
which contrast with the symmetry-free case. We also provide comprehensive
explanations of the difficulties and limitations in rigorously analyzing the
convergence time using methods that have been effective for cases without
symmetries, including Knabe's local gap threshold and Nachtergaele's martingale
methods. This suggests that a novel approach is likely necessary for
understanding the convergence time of SU$(d)$-symmetric local random circuits.
