We investigated the superconducting state of the noncentrosymmetric superconductors Li$_2$Pd$_x$Pt$_{3-x}$B with superconducting transition temperature $T_c$= 5.16(8) K ($x$=2.25), 3.56(8) K ($x=1.5$) and 2.60 K ($x=0$) by means of muon-spin rotation
($mu$SR) and specific heat experiments. The $mu$SR relaxation rate $sigma_{sc}$ was found to be constant at low temperatures for all the compounds. Data taken at different magnetic fields show that the magnetic penetration depth $lambda$ is field-independent for Li$_2$Pd$_{2.25}$Pt$_{0.75}$B and Li$_2$Pt$_{3}$B. The electronic contribution to the specific heat measured in Li$_2$Pd$_{1.5}$Pt$_{1.5}$B and Li$_2$Pt$_{3}$B increases exponentially at the lowest temperatures. These features suggest that the {it whole family} of Li$_2$Pd$_x$Pt$_{3-x}$B are single-gap s-wave superconductors across the entire doping regime.
