Weighting training data from time-series

I want to evaluate whether or not it is feasible for me to train a machine learning model using sample data that I can acquire. I have many time-series, and I am able to establish groups of ‘related’ time series – some groups are more similar than others. My hypothesis is that the history of the related time-series contains sufficient information to make some predictions about a particular time-series. I would like to make the most accurate predictions possible.

Assume I were to want to make predictions about time-series A, which is part of a group of related time series – eg. {A,B,C,D,E} – then the most relevant training data might be historic time-series for A,B,C,D and E… with the most recent examples being more relevant than the eldest. Similarly, a different time-series Z, in another group of related time-series – e.g. {Z,Y,X,W,V} might exhibit patterns relevant to prediction… but, by virtue of distinct things being measured, perhaps these examples are lower quality (from the perspective of making predictions about A) than the examples drawn from actual measurements of A and its related time-series. Perhaps there’s a cross over where more recent data about some other variables might be more relevant than elder data about the same variables.

I would like to know if there are any tools/frameworks that would help me train a machine learning model using sample data where the relevance of each training example is not uniform. Can anyone point me at a book, paper, or blog that tackles this practical problem?