High-rate codes are very attractive for quantum fault tolerance because of their low qubit overhead. Performing gates on code blocks containing multiple logical qubits presents particular challenges. In some cases, however, because the logical qubits are delocalized compared to the physical qubits, even transversal gates can perform multiple-qubit logical gates. However, we prove that there are definite limits to this approach. We show that transversal Clifford gates or code automorphisms (transversal gates plus permutations of qubits) can never achieve the full 2-qubit logical Clifford group, and fold-transversal Clifford gates cannot achieve the full 3-qubit logical Clifford group. This is work with Aranya Chakraborty, arXiv:2602.13395 [quant-ph].