On the evening of the day before Pesach, we search for Chametz and put aside what we find so that we may destroy it in some way the next morning. Before we start, we make the bracha "al biur chametz" (on the "removal/burning/destruction" of the Chametz).
But we don't destroy the Chametz until the next day. We simply put it aside (and may even consume chametz the next morning). The bracha which we made is on the completion of the act (which might, according to this answer, stretch out until the chag begins), the destruction.
As far as I have seen in siddurim, though, the concern about hefsek, an interruption, relates to the time between saying the bracha and starting the search, as per #13 here.
If the bracha actually applies to an action taken the next morning, why wouldn't there be an concern of hefsek from the moment one makes the bracha, until the moment the fire/destruction completes the act (or, as per that other answer, the holiday begins, completing the obligation)?
N.B. -- the issue might relate to the use of "al biur" according to those opinions which say that the formulation indicates imminent completion of the commandment, though it seems that some do not agree with that distinction (as per here). Also, there seems to be an idea that the search is not the completion.