The introduction of dynamic power management strategies such as clock gating and power gating in dataflow networks has been shown to provide significant energy savings when applied during idle times. However, these strategies can also degrade throughput due to shutdown and wake-up delays. Such throughput degradations might be particularly detrimental to signal processing systems that require a guaranteed throughput. As a solution, this paper first contributes a linear-program formulation for finding a periodic maximal-throughput schedule of a given so-called self-powering dataflow network where actors, realized in hardware, are allowed to go to sleep whenever not being enabled to fire. Depending on which actors are allowed to power down, tradeoffs between throughput and energy savings can be obtained. As a second contribution, we propose a mixed-integer-linear-program formulation to determine a periodic schedule that satisfies a given throughput while minimizing the overall energy per period by identifying a respective set of actors that is allowed to power down in phases of idleness and which rather not. Finally, as a third contribution, we propose a multi-objective design-space exploration strategy called "Hop and Skip" to efficiently explore the Pareto front of energy and throughput solutions. Experimental evaluations on a set of existing benchmarks and randomly generated graphs witness significant exploration time reductions over a brute-force sweep. Finally, a real-world case study is elaborated, and we report on achievable energy savings and throughputs of the related dataflow network where (a) all actors are always-active, (b) all actors are self-powered, and (c) all optimal energy and throughput tradeoff points as found by the proposed design-space exploration strategy.