Transport of pre-microRNAs (pre-miRNAs) from the nucleus to the cytoplasm. Pre-miRNAs are a ~60-70 nucleotide stem loop intermediate in miRNA production, produced by the nuclear cleavage of a primary miRNA (pri-mRNA) transcript. Pre-miRNAs are transported from the nucleus to the cytoplasm where further cleavage occurs to produce a mature miRNA product.