1. A processor, comprising: one or more processing elements, including at least a first processing element that includes one or more VA-indexed structures indexed by virtual addresses and that is configured to handle invalidation messages, the handling including: in response to determining that a first invalidation message applies to a subset of virtual addresses consisting of fewer than all virtual addresses associated with a first set of one or more translation context values, searching the VA-indexed structures in the first processing element to find any entries that correspond to a virtual address in the subset and invalidating any entries that are found; in response to determining that a second invalidation message applies to all virtual addresses associated with a second set of one or more translation context values and that no entry exists in one or more invalidation-tracking structures corresponding to the second set of one or more translation context values, bypassing searching of any of the VA-indexed structures in the first processing element to find any entries that correspond to a virtual address associated with the second set of one or more translation context values, where the one or more invalidation-tracking structures track invalidation of different sets of one or more translation context values; and in response to determining that a third invalidation message applies to all virtual addresses associated with a third set of one or more translation context values and that at least one entry exists in the one or more invalidation-tracking structures corresponding to the third set of one or more translation context values, storing invalidation information in the one or more invalidation-tracking structures to invalidate the third set of one or more translation context values and delaying searching of any of the VA-indexed structures in the first processing element to find any entries that correspond to a virtual address associated with the third set of one or more translation context values.