Two key cellular processes, namely transcription and replication, require the opening of the DNA double helix and act differently on the two DNA strands, generating different mutational patterns (mutational asymmetry) that may result, after long evolutionary time, in different nucleotide compositions on the two DNA strands (compositional asymmetry). We elaborate on the simplest model of neutral substitution rates that takes into account the strand asymmetries generated by the transcription and replication processes. Using perturbation theory, we then solve the time evolution of the DNA composition under strand-asymmetric substitution rates. In our minimal model, the compositional and substitutional asymmetries are predicted to decompose into a transcription- and a replication-associated components. The transcription-associated asymmetry increases in magnitude with transcription rate and changes sign with gene orientation while the replication-associated asymmetry is proportional to the replication fork polarity. These results are confirmed experimentally in the human genome, using substitution rates obtained by aligning the human and chimpanzee genomes using macaca and orangutan as outgroups, and replication fork polarity determined in the HeLa cell line as estimated from the derivative of the mean replication timing. When further investigating the dynamics of compositional skew evolution, we show that it is not at equilibrium yet and that its evolution is an extremely slow process with characteristic time scales of several hundred Myrs.