We have succeeded in growing high-quality single crystals of the valence-fluctuating system EuIr(2)Si(2), the divalent Eu system EuRh(2)Si(2) and the substitutional alloy Eu(Rh(1-x)Ir(x))(2)Si(2) across the range 0 < x < 1, which we characterized by means of x-ray diffraction, energy-dispersive x-ray spectroscopy, specific heat, magnetization and resistivity measurements. On increasing x, the divalent Eu ground state subsists up to x = 0.25 with a slight increase in Néel temperature, while for 0.3≤x < 0.7 a sharp hysteretic change in susceptibility and resistivity marks the first-order valence transition. For x≳0.7 the broad feature observed in the physical properties is characteristic of the continuous valence evolution beyond the critical end point of the valence transition line, and the resistivity is reminiscent of Kondo-like behaviour while the Sommerfeld coefficient indicates a mass renormalization of at least a factor of 8. The resulting phase diagram is similar to those reported for polycrystalline Eu(Pd(1-x)Au(x))(2)Si(2) and EuNi(2)(Si(1-x)Ge(x))(2), confirming its generic character for Eu systems, and markedly different to those of homologue Ce and Yb systems, which present a continuous suppression of the antiferromagnetism accompanied by a very smooth evolution of the valence. We discuss these differences and suggest them to be related to the large polarization energy of the Eu half-filled 4f shell. We further argue that the changes in the rare earth valence between RRh(2)Si(2) and RIr(2)Si(2) (R = Ce, Eu, Yb) are governed by a purely electronic effect and not by a volume effect.