We prove that near-critical percolation and dynamical percolation on the triangular lattice $η\mathbb{T}$ have a scaling limit as the mesh $η\to 0$, in the "quad-crossing" space $\mathcal{H}$ of percolation configurations introduced by Schramm and Smirnov. The proof essentially proceeds by "perturbing" the scaling limit of the critical model, using the pivotal measures studied in our earlier paper. Markovianity and conformal covariance of these new limiting objects are also established.