Adaptation of clustered regularly interspaced short palindromic repeats (CRISPR) arrays is a crucial process responsible for the unique, adaptive nature of CRISPR-Cas immune systems. The acquisition of new CRISPR spacers from mobile genetic elements has previously been studied for several types of CRISPR-Cas systems. In this study, we used a high-throughput sequencing approach to characterize CRISPR adaptation of the type V-A system from Francisella novicida and the type V-B system from Alicyclobacillus acidoterrestris. In contrast to other class 2 CRISPR-Cas systems, we found that for the type V-A and V-B systems, the Cas12 nucleases are dispensable for spacer acquisition, with only Cas1 and Cas2 (type V-A) or Cas4/1 and Cas2 (type V-B) being necessary and sufficient. Whereas the catalytic activity of Cas4 is not essential for adaptation, Cas4 activity is required for correct protospacer adjacent motif selection in both systems and for prespacer trimming in type V-A. In addition, we provide evidence for acquisition of RecBCD-produced DNA fragments by both systems, but with spacers derived from foreign DNA being incorporated preferentially over those derived from the host chromosome. Our work shows that several spacer acquisition mechanisms are conserved between diverse CRISPR-Cas systems, but also highlights unexpected nuances between similar systems that generally contribute to a bias of gaining immunity against invading genetic elements.