  Quantum critical points often arise in metals perched at the border of an
antiferromagnetic order. The recent observation of singular and dynamically
scaling charge conductivity in an antiferromagnetic quantum critical heavy
fermion metal implicates beyond-Landau quantum criticality. Here we study the
charge and spin dynamics of a Kondo destruction quantum critical point (QCP),
as realized in an SU(2)-symmetric Bose-Fermi Kondo model. We find that the
critical exponents and scaling functions of the spin and single-particle
responses of the QCP in the SU(2) case are essentially the same as those of the
large-N limit, showing that $1/N$ corrections are subleading. Building on this
insight, we demonstrate that the charge responses at the Kondo destruction QCP
are singular and obey $\omega/T$ scaling. This property persists at the Kondo
destruction QCP of the SU(2)-symmetric Kondo lattice model.
