Best way to scale across different datasets
One possible solution could be like this.

Normalize (pre-process) the dataset A such that the range of each features is within a fixed interval, e.g., between [-1, 1]. 
Train your model on the normalized set A.
Whenever you are given a new dataset like B: 

(3.1.) Normalize the new dataset such that the feature have the same range as they have in A ([-1, 1]).
(3.2) Apply your trained model (step 2) on the normalized new set (3.1).

As you have a one-to-one mapping between set B and its normalized version, then you can see what is the prediction on set B, based on predictions on normalized set B.

Note you do not need to have access to set B in advance (or such sets if they are hundreds of them). You normalize them, as soon as you are given one and you want to test your trained model on it.
I have come across a peculiar situation when preprocessing data.
Let's say I have a dataset A. I split the dataset into A_train and A_test. I fit the A_train using any of the given scalers (sci-kit learn) and transform A_test with that scaler. Now training the neural network with A_train and validating on A_test works well. No overfitting and performance is good.
Let's say I have dataset B with the same features as in A, but with different ranges of values for the features. A simple example of A and B could be Boston and Paris housing datasets respectively (This is just an analogy to say that features ranges like the cost, crime rate, etc vary significantly ). To test the performance of the above trained model on B, we transform B according to scaling attributes of A_train and then validate. This usually degrades performance, as this model is never shown the data from B.
The peculiar thing is if I fit and transform on B directly instead of using scaling attributes of A_train, the performance is a lot better. Usually, this reduces performance if I test this on A_test. In this scenario, it seems to work, although it's not right.
Since I work mostly on climate datasets, training on every dataset is not feasible. Therefore I would like to know the best way to scale such different datasets with the same features to get better performance.
Any ideas, please.
PS: I know training my model with more data can improve performance, but I am more interested in the right way of scaling. I tried removing outliers from datasets and applied QuantileTransformer, it improved performance but could be better.