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This paper introduces a framework for processing continuous, exact queries over continuous update XML streams. Instead of eagerly performing the updates on cached portions of the stream, we propagate the updates through the query evaluation pipeline, all the way to the result display, which prints the query answers. That way, the result display prints the query results continuously, replacing old results with new. The novelty of our approach is in the use of this processing framework to unblock operations and reduce buffering by letting the operations themselves embed new updates into the stream that retroactively perform the blocking parts of the operation. Based on this framework, we present novel methods for unblocking a number of important blocking/unbounded stream operations in XQuery using a small memory footprint, such as concatenation, general predicates, descendant-or-self and backward axes, and sorting.