Summary: We consider a sequence of convex integral functionals \(F_s: W^{1,p}(\Omega_s) \to\mathbb{R}\) and a sequence of weakly lower semicontinuous and generally nonintegral functionals \(G_s: W^{1,p}(\Omega_s) \to\mathbb{R}\), where \(\{\Omega_s\}\) is a sequence of domains in \(\mathbb{R}^n\) contained in a bounded domain \(\Omega\subset\mathbb{R}^n\) (\(n\geq2\)) and \(p > 1\). Along with this, we consider a sequence of closed convex sets \(V_s = \{v \in W^{1,p}(\Omega_s): v\geq K_s(v) \text{ a.e. in } \Omega_s\}\), where \(K_s\) is a mapping from the space \(W^{1,p}(\Omega_s)\) to the set of all functions defined on \(\Omega_s\). We establish conditions under which minimizers and minimum values of the functionals \(F_s + G_s\) on the sets \(V_s\) converge to a minimizer and the minimum value of a functional on the set \(V = \{v \in W^{1,p}(\Omega): v \geq K(v) \text{ a.e. in } \Omega\}\), where \(K\) is a mapping from the space \(W^{1,p}(\Omega)\) to the set of all functions defined on \(\Omega\). These conditions include, in particular, the strong connectedness of the spaces \(W^{1,p}(\Omega_s)\) with the space \(W^{1,p}(\Omega)\), the condition of exhaustion of the domain \(\Omega\) by the domains \(\Omega_s\), the \(\Gamma\)-convergence of the sequence \(\{F_s\}\) to a functional \(F: W^{1,p}(\Omega) \to \mathbb{R}\), and a certain convergence of the sequence \(\{G_s\}\) to a functional \(G: W^{1,p}(\Omega) \to \mathbb{R}\). We also assume some conditions characterizing both the internal properties of the mappings \(K_s\) and their relation to the mapping \(K\). In particular, these conditions admit the study of variational problems with irregular varying unilateral obstacles and with varying constraints combining the pointwise dependence and the functional dependence of the integral form.