The emergence of large-scale cyberinfrastructure composed of heterogeneous computing capabilities and diverse sensors and other data sources are enabling new classes of dynamic data-driven "urgent" applications. However, as the variety of data sources, and the volume and velocity of data grow, processing this data while considering the uncertainty of infrastructure and timeliness constraints of urgent application workflows can be nontrivial and presents a new set of challenges. In this paper, we use an application workflow that monitors and manage the air quality impacts of remote wildfires to illustrate how the R-Pulsar programming system, leveraging the SAGE and WIFIRE platforms, can enable urgent analytics across the computing continuum. R-Pulsar supports urgent data-processing pipelines that tradeoff the content of data, cost of computation, and urgency of the results to support such workflows. We also discuss research challenges associated with programming urgent application workflows and managing resources in an autonomic manner.