A data processing system (2) including one or more transaction buffers (16, 18, 20) storing address translation data executes translation buffer invalidation instructions TLBI within respective address translation contexts VMID, ASID, X. Translation buffer invalidation signals generated as a consequence of execution of the translation buffer invalidation instructions are broadcast to respective translation buffers and include signals which specify the address translation context of the translation buffer invalidation instruction that was executed. This address translation context specified within the translation buffer invalidation signals is used to gate whether or not those translation buffer invalidation signals when received by translation buffers which are potential targets for the invalidation are or are not flushed. The address translation context data provided within the translation buffer invalidation signals may also be used to control whether or not local memory transactions for a local transactional memory access are or are not aborted upon receipt of the translation buffer invalidation signals.