The kernel of a binary code containing the zero word partitions the binary coordinates into blocks, two coordinates lying in the same block when every kernel word takes the same value in both, and the \emph{kernel-block rank profile} is the multiset of the dimensions of its linear span punctured on those blocks. For the family $H^{t_1,t_2,t_3}$ of $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard codes, this invariant is known explicitly and gives a complete classification of the family. In this paper, we compute it for the $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard families with which those codes are compared, and we prove that it is constant for every nonlinear $\mathbb{Z}_2\mathbb{Z}_4$-linear Hadamard code and for every nonlinear $\mathbb{Z}_{2^s}$-linear Hadamard code $\bar H^{a_1,\dots,a_s}$ with $s\geq2$. The second statement is obtained without any rank formula, by exhibiting coordinate permutations that preserve the code and act transitively on its kernel blocks, which makes the argument uniform in $s$. We also prove a descent theorem: if $\bar H^{a_1,\dots,a_s}$ is nonlinear and $a_1\geq2$, then the code punctured on one kernel block is the $\mathbb{Z}_{2^{s-1}}$-linear Hadamard code $\bar H^{a_1,\dots,a_{s-1}}$. Consequently, the constant local rank equals $rank(\bar H^{a_1,\dots,a_{s-1}})$ and is at least $t-κ+2$, where $2^t$ is the length and $κ$ the kernel dimension. These results separate every nonlinear $\mathbb{Z}_2\mathbb{Z}_4\mathbb{Z}_8$-linear Hadamard code from every $\mathbb{Z}_4$-linear, $\mathbb{Z}_2\mathbb{Z}_4$-linear and $\mathbb{Z}_{2^s}$-linear Hadamard code of the same length, except for the single infinite family $H^{1,1,t-4}$ and $\bar H^{2,0,t-5}$ with $t\geq5$. The members of this infinite family agree in the rank, kernel dimension and the kernel-block rank profile, but a two-block refinement separates them.

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