Genetic variants identified through genome-wide association studies (GWAS) are typically non-coding, exerting small regulatory effects on downstream genes. However, which downstream genes are ultimately impacted and how they confer risk remains mostly unclear. By contrast, variants that cause rare Mendelian diseases are often coding and have a more direct impact on disease development. Here we demonstrate that common and rare genetic diseases can be linked by studying how common disease-associated variants influence gene co-expression in 57 different tissues and cell types. We implemented this method in a framework called Downstreamer and applied it to 88 GWAS traits. We find that predicted downstream "genes" are enriched with Mendelian disease genes, e.g. key genes for height are enriched for genes that cause skeletal abnormalities and Ehlers-Danlos syndromes. 78% of these key genes are located outside of GWAS loci, suggesting that they result from complex trans regulation rather than being impacted by disease-associated variants in cis. Based on our findings, we discuss the challenges in reconstructing gene regulatory networks and provide a roadmap to improve the identification of these highly connected genes linked to common traits and diseases.