  This is the first of a series of papers devoted to a thorough analysis of the
class of gradient flows in a metric space $(X,\mathsf{d})$ that can be
characterized by Evolution Variational Inequalities. We present new results
concerning the structural properties of solutions to the $\mathrm{EVI}$
formulation, such as contraction, regularity, asymptotic expansion, precise
energy identity, stability, asymptotic behaviour and their link with the
geodesic convexity of the driving functional. Under the crucial assumption of
the existence of an $\mathrm{EVI}$ gradient flow, we will also prove two main
results: the equivalence with the De Giorgi variational characterization of
curves of maximal slope and the convergence of the Minimizing Movement-JKO
scheme to the $\mathrm{EVI}$ gradient flow, with an explicit and uniform error
estimate of order $1/2$ with respect to the step size, independent of any
geometric hypothesis (such as upper or lower curvature bounds) on $\mathsf{d}$.
In order to avoid any compactness assumption, we will also introduce a suitable
relaxation of the Minimizing Movement algorithm obtained by the Ekeland
variational principle, and we will prove its uniform convergence as well.
