Condensation by phase separation has recently emerged as a mechanism underlying many nuclear compartments essential for cellular functions. Nuclear condensates enrich nucleic acids and proteins, localize to specific genomic regions, and often promote gene expression. How diverse properties of nuclear condensates are shaped by gene organization and activity is poorly understood. Here, we develop a physics-based model to interrogate how spatially-varying transcription activity impacts condensate properties and dynamics. Our model predicts that spatial clustering of active genes can enable precise localization and de novo nucleation of condensates. Strong clustering and high activity results in aspherical condensate morphologies. Condensates can flow towards distant gene clusters and competition between multiple clusters lead to stretched morphologies and activity-dependent repositioning. Overall, our model predicts and recapitulates morphological and dynamical features of diverse nuclear condensates and offers a unified mechanistic framework to study the interplay between non-equilibrium processes, spatially-varying transcription, and multicomponent condensates in cell biology.