The affinity constants, k and k, of the 1879 paper can now be recognised as rate constants. The equilibrium constant, K, was derived by setting the rates of forward and backward reactions to be equal. This also meant that the chemical affinities for the forward and backward reactions are equal. The resultant expression
is correct even from the modern perspective, apart from the use of concentrations instead of activities (the concept of chemical activity was developed by Josiah Willard Gibbs, in the 1870s, but was not widely known in Europe until the 1890s). The derivation from the reaction rate expressions is no longer considered to be valid. Nevertheless, Guldberg and Waage were on the right track when they suggested that the driving force for both forward and backward reactions is equal when the mixture is at equilibrium. The term they used for this force was chemical affinity. Today the expression for the equilibrium constant is derived by setting the chemical potential of forward and backward reactions to be equal. The generalisation of the law of mass action, in terms of affinity, to equilibria of arbitrary stoichiometry was a bold and correct conjecture.
The hypothesis that reaction rate is proportional to reactant concentrations is, strictly speaking, only true for elementary reactions (reactions with a single mechanistic step), but the empirical rate expression
is also applicable to second order reactions that may not be concerted reactions. Guldberg and Waage were fortunate in that reactions such as ester formation and hydrolysis, on which they originally based their theory, do indeed follow this rate expression.
In general many reactions occur with the formation of reactive intermediates, and/or through parallel reaction pathways. However, all reactions can be represented as a series of elementary reactions and, if the mechanism is known in detail, the rate equation for each individual step is given by the  expression so that the overall rate equation can be derived from the individual steps. When this is done the equilibrium constant is obtained correctly from the rate equations for forward and backward reaction rates.
In biochemistry, there has been significant interest in the appropriate mathematical model for chemical reactions occurring in the intracellular medium.  This is in contrast to the initial work done on chemical kinetics, which was in simplified systems where reactants were in a relatively dilute, pH-buffered, aqueous solution.  In more complex environments, where bound particles may be prevented from disassociation by their surroundings, or diffusion is slow or anomalous, the model of mass action does not always describe the behavior of the reaction kinetics accurately.  Several attempts have been made to modify the mass action model, but consensus has yet to be reached.  Popular modifications replace the rate constants with functions of time and concentration.  As an alternative to these mathematical constructs, one school of thought is that the mass action model can be valid in intracellular environments under certain conditions, but with different rates than would be found in a dilute, simple environment .
The fact that Guldberg and Waage developed their concepts in steps from 1864 to 1867 and 1879 has resulted in much confusion in the literature as to which equation the law of mass action refers. It has been a source of some textbook errors.  Thus, today the "law of mass action" sometimes refers to the (correct) equilibrium constant formula,
and at other times to the (usually incorrect)  rate formula.