Collective tunneling is a ubiquitous phenomenon in finite-size spin clusters that shows up in systems as diverse as molecular magnets or spin clusters adsorbed at surfaces. The basic problem we explore is to understand how small flipping terms can cooperate to flip a large spin to the opposite direction or a cluster of interacting elementary Ising spins into the time-reversed state. These high-order processes will involve at least two channels, a single spin-flip channel due to a transverse field and a two-spin flip channel due to exchange or other pairwise interactions or due to single-ion anisotropies. In view of the complexity of high-order perturbation theory, nonperturbative approaches based on large-spin path integrals were developed when this problem was first addressed in the context of single spin models. In the present paper, we show that high-order perturbation theory can in fact be formulated and evaluated with the help of simple recurrence relations, leading to a compact theory of tunneling in macroscopic spins, in one-dimensional clusters, as well as in small higher-dimensional clusters. This is demonstrated explicitly in the case of the Ising model with a transverse field and transverse exchange, and in the case of macroscopic spins with uniaxial anisotropy. Our approach provides a transparent theory of level crossings, where the tunneling between time-reversed configurations vanishes as a function of the external field. Those crossings result from the destructive quantum interferences between competing flipping channels. Destructive interferences are expected to be present as soon as the two-spin flip channels have an overall positive amplitude and thus compete with the intrinsically negative second-order processes due to the transverse field. Our theory consistently predicts N crossings in chains of N Ising spins and 2S crossings in single spins of magnitude S and yields explicit analytical formulas for the level crossings of open chains and macroscopic spins. Disorder can be easily implemented in this perturbative formalism. Leading disorder effects can be treated analytically for spin rings. We find that at the smallest transverse field crossing the suppression of tunneling is most robust with respect to disorder and fluctuations in the parameters. We briefly discuss the implications of our findings for the use of realistic spin clusters on surfaces to store information.