By using Monte Carlo (MC) techniques, we study the effect of the transient mobility on the irreversible deposition of spherical particles on two-dimensional substrates. The mobility is introduced by the capability of the incoming particles to roll on the flat surface a distance R before coming to rest or, in the case of falling on top of an already adsorbed particle, roll over until it contacts the surface. Depending on the deposition mechanism three models of irreversible adsorption with transient mobility are proposed. The kinetics of the process are studied by the time behavior of coverage rho(t). The asymptotic regime rho(infinity)-rho(t) appears as specially interesting and behaves as a function of the degrees of freedom in the motion of the depositing particles going from power law to exponential dependence. The jamming state is characterized through the behavior of the jamming coverage rho(infinity) and the correlation function g(r).