Pursuing non‐equilibrium chemistry with (bio)molecules is of utmost importance for the design of life‐like dynamic materials that emerge in a constant flux of energy. Herein, we explore spatial localization of a dissipative self‐assembly of biocondensate (DNA‐histone) via passing chemical fuel (histone) and one fuel‐degrading agent (trypsin) through two arms of the Y‐shaped microfluidic chip. In this case, continuous supply of fuel and fuel‐degrading agent results self‐assembly of biocondensate, maintaining a non‐equilibrium steady state (NESS). We find in the presence of gradient of dissipating conditions, the formation zone of biocondensate drifts towards fuel‐rich zone (away from dissipating zone). In absence of fuel‐degrading agent, diffusive transport of free DNA towards histone channel (perpendicular to advection) is restricted as it formed much larger micron‐sized biocondensate at the center of the channel (the meeting point of two flows). However, this sidewise DNA diffusion is operative in the presence of fuel‐degrading agent and therefore, the formation zone shifted to histone‐rich zone. Furthermore, we demonstrate that in the presence of trypsin, catalytic DNA’s peroxidase reactivity can be moved to histone‐rich region. Transposition of self‐assembly process in a gradient of dissipative conditions will be of importance in the development of spatially‐controlled chemistry, reaction‐diffusion processes.