Convolutional Neural Networks (CNNs) are the backbone of many deep learning methods, but optimizing them remains computationally expensive. To address this, we explore multiscale training frameworks and mathematically identify key challenges, particularly when dealing with noisy inputs. Our analysis reveals that in the presence of noise, the gradient of standard CNNs in multiscale training may fail to converge as the mesh-size approaches to 0, undermining the optimization process. This insight drives the development of Mesh-Free Convolutions (MFCs), which are independent of input scale and avoid the pitfalls of traditional convolution kernels. We demonstrate that MFCs, with their robust gradient behavior, ensure convergence even with noisy inputs, enabling more efficient neural network optimization in multiscale settings. To validate the generality and effectiveness of our multiscale training approach, we show that (i) MFCs can theoretically deliver substantial computational speedups without sacrificing performance in practice, and (ii) standard convolutions benefit from our multiscale training framework in practice.