We study the expectation values of observables and correlation functions at long times after a global quantum quench. Our focus is on metallic (“gapless”) fermionic many-body models and small quenches. The system is prepared in an eigenstate of an initial Hamiltonian, and the time evolution is performed with a final Hamiltonian which differs from the initial one in the value of one global parameter. We first derive general relations between time-averaged expectation values of observables as well as correlation functions and those obtained in an eigenstate of the final Hamiltonian. Our results are valid to linear and quadratic order in the quench parameter g and generalize prior insights in several essential ways. This allows us to develop a phenomenology for the thermalization of local quantities up to a given order in g. Our phenomenology is put to a test in several case studies of one-dimensional models representative of four distinct classes of Hamiltonians: quadratic ones, effectively quadratic ones, those characterized by an extensive set of (quasi-) local integrals of motion, and those for which no such set is known (and believed to be nonexistent). We show that for each of these models, all observables and correlation functions thermalize to linear order in g. The more local a given quantity, the longer the linear behavior prevails when increasing g. Typical local correlation functions and observables for which the term O(g) vanishes thermalize even to order g2. Our results show that lowest-order thermalization of local observables is an ubiquitous phenomenon even in models with extensive sets of integrals of motion.