A new CMS tracker detector will be installed for operation at the High Luminosity LHC (HL-LHC). This detector comprises modules with two closely spaced parallel sensor plates and front-end ASICs capable of transmitting tracking information to the CMS Level-1 (L1) trigger at the 40 MHz beam crossing rate. The inclusion of tracking information in the L1 trigger decision will be essential for selecting events of interest efficiently at the HL-LHC. The CMS Binary Chip (CBC) has been designed to read out and correlate hits from pairs of tracker sensors, forming so-called track stubs. For the first time, a prototype irradiated module and a full-sized module, both equipped with the version 2 of the CBC, have been operated in test beam facilities. The efficiency of the stub finding logic of the modules for various angles of incidence has been studied. The ability of the modules to reject tracks with transverse momentum less than 2 GeV has been demonstrated. For modules built with irradiated sensors, no significant drop in the stub finding performance has been observed. Results from the beam tests are described in this paper.