It is universally true in ecological communities, terrestrial or aquatic, temperate or tropical, that some species are very abundant, others are moderately common, and the majority are rare. Likewise, eukaryotic genomes also contain classes or “species” of genetic elements that vary greatly in abundance: DNA transposons, retrotransposons, satellite sequences, simple repeats and their less abundant functional sequences such as RNA or genes. Are the patterns of relative species abundance and diversity similar among ecological communities and genomes? Previous dynamical models of genomic diversity have focused on the selective forces shaping the abundance and diversity of transposable elements (TEs). However, ideally, models of genome dynamics should consider not only TEs, but also the diversity of all genetic classes or “species” populating eukaryotic genomes. Here, in an analysis of the diversity and abundance of genetic elements in >500 eukaryotic chromosomes, we show that the patterns are consistent with a neutral hypothesis of genome assembly in virtually all chromosomes tested. The distributions of relative abundance of genetic elements are quite precisely predicted by the dynamics of an ecological model for which the principle of functional equivalence is the main assumption. We hypothesize that at large temporal scales an overarching neutral or nearly neutral process governs the evolution of abundance and diversity of genetic elements in eukaryotic genomes.