How should one check an instant-equilibration algorithm for conservation? We have a system of equations that describes the equilibration between several phases $A, B, ...$.  If we define the total concentration $T = A + B + ...$, then $\partial_t T=0$ must hold.

We are further assuming equilibrium at all times, so we have devised an algorithm for this that transforms $A\rightarrow A'$, $B\rightarrow B'$ etc.  Finally, the total concentration must be conserved, so we must check if $T'=T$.  But should one also check if $\partial_t T'=0$, i.e. the conservation equation with the equilibrated variables?

Does it make sense to test both $T'-T=0$ and $\partial_t T'=0$?
I would think one of those conditions would always be sufficient. If you have shown that $T'-T=0$ for all $t$ and that $\partial_tT=0$, then  $\partial_tT'=0$ must be true by default. That is $\partial_tT=0$ shows that $T$ is a constant and if $T'-\text{constant}=0$ for all $t$, then $T'$ is also a constant.