The non–steady state solidification of RE123 (RE = Y, Nd) superconductive oxides was investigated by using an undercooling growth method. In both the Y– and Nd–systems, RE123 crystals grew steadily in the initial growth stage, but the growth rates decreased gradually in the later stage under the constant growth temperature. In the case of Nd–system, the substitution ratio of Hd/Ba in the grown Nd123 crystal was found to be gradually changed. The liquid compositions revealed that the liquid composition was close to the ternary equilibrium point after the growth rates decreased. This non-steady growth of Nd123 crystal is intrinsically caused since the composition of grown Nd123 solid solution is not on the 422–123 line and the residual liquid composition gradually shifts to the ternary equilibrium point to compensate for the mass balance during 123 growth. On the other hand, in the case of the Y–system, the evaluation of the volume fraction of Y211 particles showed that the volume fractions in the grown crystal were lower than those expected from the initial composition. On the contrary, those in liquid were higher and had similar valuesirrespective of the initial compositions and growth conditions. This accumulation of 211 phase particles near the 123/liquid interface would reduce the diffusion of the solute elements, which have the partition ratio of less than unity, away from the 123 interface. Therefore, such a large volumefraction would enhance the compositional shift close to the growth interface and cause locally a final transient to decrease the growth rate of the 123 crystal, if there is a compositional shift from the line connecting 123 and 211 compositions.