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We prove that Fomin's generalization of Lindstr\"om's lemma for paths on acyclic directed graphs to walks on general directed graphs also generalizes a theorem of Stembridge in the same way. Moreover, we show that whenever a family of operations satisfies a Lindstr\"om-type determinant relation, a related family of operations satisfies a Stembridge-type Pfaffian relation. We give example applications to Kenyon and Wilson's work on groves and to Talaska's work on alternating flows.

A Principle for Converting Lindström-Type Lemmas to Stembridge-Type Theorems, with Applications to Walks, Groves, and Alternating Flows