Rapidity-odd directed-flow measurements at midrapidity are presented for Lambda, (Lambda) over bar, K-+/-, K-s(0), and phi at root s(NN) = 7.7, 11.5, 14.5, 19.6, 27, 39, 62.4, and 200 GeV in Au + Au collisions recorded by the Solenoidal Tracker detector at the Relativistic Heavy Ion Collider. These measurements greatly expand the scope of data available to constrain models with differing prescriptions for the equation of state of quantum chromodynamics. Results show good sensitivity for testing a picture where flow is assumed to be imposed before hadron formation and the observed particles are assumed to form via coalescence of constituent quarks. The pattern of departure from a coalescence-inspired sum rule can be a valuable new tool for probing the collision dynamics.