Understanding the adsorption and mobility of metal organic framework (MOF)-supported metal nanoclusters is critical to the development of these catalytic materials. We present the first theoretical investigation of Au-, Pd-, and AuPd-supported clusters in a MOF, namely MOF-74. We combine density functional theory (DFT) calculations with a genetic algorithm (GA) to reliably predict the structure of the adsorbed clusters. This approach allows comparison of hundreds of adsorbed configurations for each cluster. From the investigation of Au-8, Pd-8, and Au4Pd4, we find that the organic part of the MOF is just as important for nanocluster adsorption as open Zn or Mg metal sites. Using the large number of clusters generated by the GA, we developed a systematic method for predicting the mobility of adsorbed clusters. Through the investigation of diffusion paths a relationship between the cluster's adsorption energy and diffusion barrier is established, confirming that Au clusters are highly mobile in the MOF-74 framework and Pd clusters are less mobile.