  We develop tools for performing effective field theory (EFT) calculations in
a manifestly gauge-covariant fashion. We clarify how functional methods account
for one-loop diagrams resulting from the exchange of both heavy and light
fields, as some confusion has recently arisen in the literature. To efficiently
evaluate functional traces containing these "mixed" one-loop terms, we develop
a new covariant derivative expansion (CDE) technique that is capable of
evaluating a much wider class of traces than previous methods. The technique is
detailed in an appendix, so that it can be read independently from the rest of
this work. We review the well-known matching procedure to one-loop order with
functional methods. What we add to this story is showing how to isolate
one-loop terms coming from diagrams involving only heavy propagators from
diagrams with mixed heavy and light propagators. This is done using a non-local
effective action, which physically connects to the notion of "integrating out"
heavy fields. Lastly, we show how to use a CDE to do running analyses in EFTs,
\textit{i.e.} to obtain the anomalous dimension matrix. We demonstrate the
methodologies by several explicit example calculations.
