The authors consider four different non-interacting particle systems on the line. There are \(n\) particles, each of which attempts to move right in turn by a number of steps given by either a geometric or Bernoulli distribution; however, they are forbidden to pass, either by any particle trying to pass the particle to its right pushing that particle along with it, or by stopping at the position of that particle. The transition kernels for all four models are intertwined with a Karlin-McGregor-type kernel [S. Karlin and J. McGregor, Pac. J. Math.9, 1141–1164 (1959; Zbl 0092.34503)]; the kernels inherit the determinantal structure from the Karlin-McGregor formula, with a form similar to Schütz’s kernel for the totally asymmetric exclusion process [G. M. Schütz, J. Stat.Phys.88, No.1–2, 427–445 (1997; Zbl 0945.82508)]. The proofs involve using the Robinson-Schensted-Knuth correspondence to constuct a bijection between the motions of the particles and pairs of semistandard Young tableaux; the determinants arise from the formulas for the enumeration of tableaux.