Xist RNA initiates X inactivation as it spreads in cis across the chromosome. Here, we reveal a biophysical basis for its cis-limited diffusion. Xist RNA and HNRNPK together drive a liquid-liquid phase separation (LLPS) that encapsulates the chromosome. HNRNPK droplets pull on Xist and internalize the RNA. Once internalized, Xist induces a further phase transition and "softens" the HNRNPK droplet. Xist alters the condensate's deformability, adhesiveness, and wetting properties in vitro. Other Xist-interacting proteins are internalized and entrapped within the droplet, resulting in a concentration of Xist and protein partners within the condensate. We attribute LLPS to HNRNPK's RGG and Xist's repeat B (RepB) motifs. Mutating these motifs causes Xist diffusion, disrupts polycomb recruitment, and precludes the required mixing of chromosomal compartments for Xist's migration. Thus, we hypothesize that phase transitions in HNRNPK condensates allow Xist to locally concentrate silencing factors and to spread through internal channels of the HNRNPK-encapsulated chromosome.