We consider the sequential allocation of $m$ balls (jobs) into $n$ bins (servers) by allowing each ball to choose from some bins sampled uniformly at random. The goal is to maintain a small gap between the maximum load and the average load. In this paper, we present a general framework that allows us to analyze various allocation processes that slightly prefer allocating into underloaded, as opposed to overloaded bins. Our analysis covers several natural instances of processes, including: The Caching process (a.k.a. memory protocol) as studied by Mitzenmacher, Prabhakar and Shah (2002): At each round we only take one bin sample, but we also have access to a cache in which the most recently used bin is stored. We place the ball into the least loaded of the two. The Packing process: At each round we only take one bin sample. If the load is below some threshold (e.g., the average load), then we place as many balls until the threshold is reached; otherwise, we place only one ball. The Twinning process: At each round, we only take one bin sample. If the load is below some threshold, then we place two balls; otherwise, we place only one ball. The Thinning process as recently studied by Feldheim and Gurel-Gurevich (2021): At each round, we first take one bin sample. If its load is below some threshold, we place one ball; otherwise, we place one ball into a $\textit{second}$ bin sample. As we demonstrate, our general framework implies for all these processes a gap of $\mathcal{O}(\log n)$ between the maximum load and average load, even when an arbitrary number of balls $m \geq n$ are allocated (heavily loaded case). Our analysis is inspired by a previous work of Peres, Talwar and Wieder (2010) for the $(1+β)$-process, however here we rely on the interplay between different potential functions to prove stabilization.