The increasing size and complexity of modern deep neural networks (DNNs) pose significant challenges for on-device inference on mobile GPUs, with limited memory and computational resources. Existing DNN acceleration frameworks primarily deploy a weight preloading strategy, where all model parameters are loaded into memory before execution on mobile GPUs. We posit that this approach is not adequate for modern DNN workloads that comprise very large model(s) and possibly execution of several distinct models in succession. In this work, we introduce FlashMem, a memory streaming framework designed to efficiently execute large-scale modern DNNs and multi-DNN workloads while minimizing memory consumption and reducing inference latency. Instead of fully preloading weights, FlashMem statically determines model loading schedules and dynamically streams them on demand, leveraging 2.5D texture memory to minimize data transformations and improve execution efficiency. Experimental results on 11 models demonstrate that FlashMem achieves 2.0x to 8.4x memory reduction and 1.7x to 75.0x speedup compared to existing frameworks, enabling efficient execution of large-scale models and multi-DNN support on resource-constrained mobile GPUs.