This paper presents the crossing scheme (X-scheme) for improving the performance of deep neural network (DNN)-based music source separation (MSS) with almost no increasing calculation cost. It consists of three components: (i) multi-domain loss (MDL), (ii) bridging operation, which couples the individual instrument networks, and (iii) combination loss (CL). MDL enables the taking advantage of the frequency- and time-domain representations of audio signals. We modify the target network, i.e., the network architecture of the original DNN-based MSS, by adding bridging paths for each output instrument to share their information. MDL is then applied to the combinations of the output sources as well as each independent source; hence, we called it CL. MDL and CL can easily be applied to many DNN-based separation methods as they are merely loss functions that are only used during training and do not affect the inference step. Bridging operation does not increase the number of learnable parameters in the network. Experimental results showed that the validity of Open-Unmix (UMX), densely connected dilated DenseNet (D3Net) and convolutional time-domain audio separation network (Conv-TasNet) extended with our X-scheme, respectively called X-UMX, X-D3Net and X-Conv-TasNet, by comparing them with their original versions. We also verified the effectiveness of X-scheme in a large-scale data regime, showing its generality with respect to data size. X-UMX Large (X-UMXL), which was trained on large-scale internal data and used in our experiments, is newly available at https://github.com/asteroid-team/asteroid/tree/master/egs/musdb18/X-UMX.