Despite insertions and deletions being the most common structural variants (SVs) found across genomes, not much is known about how much these SVs vary within populations and between closely related species, nor their significance in evolution. To address these questions, we characterized the evolution of indel SVs using genome assemblies of three closely related Heliconius butterfly species. Over the relatively short evolutionary timescales investigated, up to 18.0% of the genome was composed of indels between two haplotypes of an individual Heliconius charithonia butterfly and up to 62.7% included lineage-specific SVs between the genomes of the most distant species (11 Mya). Lineage-specific sequences were mostly characterized as transposable elements (TEs) inserted at random throughout the genome and their overall distribution was similarly affected by linked selection as single nucleotide substitutions. Using chromatin accessibility profiles (i.e., ATAC-seq) of head tissue in caterpillars to identify sequences with potential cis -regulatory function, we found that out of the 31,066 identified differences in chromatin accessibility between species, 30.4% were within lineage-specific SVs and 9.4% were characterized as TE insertions. These TE insertions were localized closer to gene transcription start sites than expected at random and were enriched for sites with significant resemblance to several transcription factor binding sites with known function in neuron development in Drosophila . We also identified 24 TE insertions with head-specific chromatin accessibility. Our results show high rates of structural genome evolution that were previously overlooked in comparative genomic studies and suggest a high potential for structural variation to serve as raw material for adaptive evolution.