We present a modeling framework for the dynamics of a population of individuals with a continuum of traits, who compete for resources and exchange horizontally (transfer) an otherwise vertically inherited trait. Competition influences individual demographics, affecting population size, which feeds back on the dynamics of transfer. This feedback is captured with a stochastic individual-based model, from which a deterministic approximation for large populations is derived. The limiting process is the solution of a non linear integro-differential equation. When there are only two different traits, the equation reduces to a non-standard two-dimensional dynamical system. We show how crucial the forms of the transfer rates are for the long-term behavior of its solutions. For density-dependent transfer rates, the equilibria are those of Lotka-Volterra systems, but horizontal transfer may revert the change of evolution. Frequency-dependence or more general transfert rates reveal new phase diagrams. The interaction between horizontal transfer and competition makes for example possible the stable (or bi-stable) polymorphic maintenance of deleterious traits (including costly plasmids). For an initially rare trait, we describe the dynamics of invasion and fixation and compute the invasion probabilities. Horizontal transfer can have a major impact on the distribution of the mutational effects that are fixed. This allows us to consider then the impact of horizontal transfer on evolution. Our model provides a basis for a general theory of the influence of horizontal transfer on eco-evolutionary dynamics and adaptation.