Ramping neuronal activity refers to spiking activity with a rate that increases quasi-linearly over time. It has been observed in multiple cortical areas and is correlated with evidence accumulation processes or timing. In this work, we investigated the downstream effect of ramping neuronal activity through synapses that display short-term facilitation (STF) or depression (STD). We obtained an analytical result for a synapse driven by deterministic linear ramping input that exhibits pure STF or STD and numerically investigated the general case when a synapse displays both STF and STD. We show that the analytical deterministic solution gives an accurate description of the averaging synaptic activation of many inputs converging onto a postsynaptic neuron, even when fluctuations in the ramping input are strong. Activation of a synapse with STF shows an initial cubical increase with time, followed by a linear ramping similar to a synapse without STF. Activation of a synapse with STD grows in time to a maximum before falling and reaching a plateau, and this steady state is independent of the slope of the ramping input. For a synapse displaying both STF and STD, an increase in the depression time constant from a value much smaller than the facilitation time constant to a value much larger than leads to a transition from facilitation dominance to depression dominance. Therefore, our work provides insights into the impact of ramping neuronal activity on downstream neurons through synapses that display short-term plasticity. In a perceptual decision-making process, ramping activity has been observed in the parietal and prefrontal cortices, with a slope that decreases with task difficulty. Our work predicts that neurons downstream from such a decision circuit could instead display a firing plateau independent of the task difficulty, provided that the synaptic connection is endowed with short-term depression.