We present measurements of the elliptic flow (v(2)) as a function of transverse momentum (p(T)), pseudorapidity (eta), and centrality in d + Au collisions at root s(NN) = 200, 62.4, 39, and 19.6 GeV. The beam-energy scan of d + Au collisions provides a testing ground for the onset of flow signatures in small collision systems. We measure a nonzero v(2) signal at all four collision energies, which, at midrapidity and low p(T), is consistent with predictions from viscous hydrodynamic models. Comparisons with calculations from parton transport models (based on the AMPT Monte Carlo generator) show good agreement with the data at midrapidity to forward (d-going) rapidities and low p(T). At backward (Au-going) rapidities and p(T) > 1.5GeV/c, the data diverges from AMPT calculations of v(2) relative to the initial geometry, indicating the possible dominance of nongeometry related correlations, referred to as nonflow. We also present measurements of the charged-particle multiplicity (d N-ch/d eta) as a function of eta in central d + Au collisions at the same energies. We find that in d + Au collisions at root s(NN) = 200 GeV the v(2) scales with d N-ch/d eta over all eta in the PHENIX acceptance. At root s(NN) = 62.4, and 39 GeV, v(2) scales with d N-ch/d eta at midrapidity and forward rapidity, but falls off at backward rapidity. This departure from the d N-ch/d eta scaling may be a further indication of nonflow effects dominating at backward rapidity.