A generalized random sequential adsorption (RSA) approach is developed by taking into account diffusion, particle/wall hydrodynamic interactions as well as external forces (e.g., gravity). In analogy to the previous concept of Pagonabarraga and Rubi＇ [Physica A 188, 553 (1992)] the position dependent available surface function <(Phi)over bar>(z, theta) is introduced. Using this definition, constitutive expressions for the adsorption flux are formulated which represent the generalization of previous models, including the widely used Langmuirian kinetic approach. It is shown that the overall available surface function Phi(Delta, theta) plays the crucial role in these expressions. It represents the net probability of transferring a particle from the arbitrary distance Delta to the interface for a given surface coverage. Explicit expressions in the form of definite integrals are formulated which enable one to calculate the <(Phi)over bar>(Delta, theta) function in terms of the Phi(z, theta). In the case of hard spheres, Phi(z, theta) is calculated up to the second order of the surface coverage theta using geometrical arguments. The effect of an external force gravity is characterized by the dimensionless radius of particles R*, where R* --> -infinity corresponds to the purely ballistic case, R* --> -infinity to the diffusion RSA, and R* --> -infinity reflects the case of infinite gravity acting outwards from the surface. Using these expressions, the overall <(Phi)over bar>(Delta, theta) function is also calculated. It is found that the RSA available surface function is not recovered for R* = 0 as expected, but for R* --> -infinity. The transition from the R* = 0 to the ballistic case (R* = infinity) is analyzed. Unexpectedly, it is found that for R* = 1 the second order terminal the coverage expansion of <(Phi)over bar>(Delta, theta) appears negative which seems an entirely new result. It is also deduced that in the case of an energy barrier, whose extension is much smaller than the particle dimension, the adsorption process can well be characterized for R* = 0 in terms of the classical RSA model. This can be explained by the fact that for a high energy barrier the adsorbing particles could randomize over the deposition plane before crossing the barrier and adsorbing irreversibly.