We present an extensive study of the structural, magnetic and thermodynamic properties of high-quality monocrystals of two heterometallic oxyborates from the ludwigite family: Cu₂GaBO₅ and Cu₂AlBO₅ in the temperature range above 2 K. The distinctive feature of the investigated structures is the selective distribution of Cu and Ga/Al cations. The unit cell of Cu₂GaBO₅ and Cu₂AlBO₅ contains four nonequivalent crystallographic sites of metal ions. Two sites in the structure of Cu₂GaBO₅ are fully occupied by Cu atoms which form the quasi one-dimensional chains along the a-axis. For Cu₂AlBO₅ all sites are partially occupied by Al and Cu atoms. The joint analysis of low-temperature data on magnetic susceptibility and magnetic contribution to the specific heat showed that Cu₂AlBO₅ and Cu₂GaBO₅ exhibit an antiferromagnetic transition at T_N=2.4 and 4.1 K, respectively. The magnetic response below T_N can be represented as the sum of two contributions: the ordered component and the paramagnetic contribution from defects of the same relative weight. It is shown that the external magnetic field above 2.5 T and 2.8 T for Cu₂GaBO₅ and Cu₂AlBO₅, respectively, leads to a broadening of the magnetic phase transition indicating suppression of the long-range antiferromagnetic order.