We probe the history of rapidly radiating Heliconius butterflies by means of 20 new genome assemblies and employ them to investigate the genomic architecture of gene flow among lineages. By developing a test to distinguish incomplete lineage sorting from introgression, we demonstrate that histories of loci that differ from the species tree arose mostly through introgression. Moreover, these loci are underrepresented in low recombination and gene-rich regions, consistent with the purging of introgressed alleles tightly linked with incompatibility loci. Additionally, our analysis identifies an inversion that captures a color pattern switch locus which was transferred between lineages via introgression and is convergent with a similar rearrangement in another part of the genus. This analysis of multiple de novo genome sequences enables an improved understanding of the importance of introgression and selective processes in adaptive radiation.