We have constructed conserved $N=2$ higher-spin supercurrents for an arbitrary integer spin through the unconstrained superfield formulation of $4D,N=2$ massless higher-spin theories in the harmonic superspace. The obtained supercurrents are gauge-invariant and determine consistent cubic interactions in the gauge superfield × supercurrent form for massless spin $s_1$ and two massless spin $s_2$ $N=2$ supermultiplets. Such interactions and $N=2$ supercurrents exist only for $s_1\ge 2s_2$. These supercurrents are the $N=2$ supersymmetric extension of Berends-Burgers-van Dam higher-spin currents and generalize the linearized Bel-Robinson tensor. The relevant $N=2$ supercurrents can be considered as the descendants of the $N=2$ principle supercurrent, which has a simple universal structure. An important feature of our outcomes is that, ultimately, all $N=2$ supercurrents are built from the $N=2$ superfield strengths, which are constructed from the unconstrained analytical higher-spin prepotentials.