So far, I've had 4 important relationships while being poly - my ex-husband, my current husband Adam, and my partners Brian and Greg.
I got divorced from my ex, and stayed with Adam a couple months while I waited for my mother to move from her place so I could move in there. This happened at about 6 months into our relationship, and my unsolicited advice is that if there is another option, don't move in together. Lots of reasons, but it drove us up a relationship escalator when it probably wasn't the best idea. This was how Adam supported me - offering me a place to stay.
Now my partners Brian and Greg, both of them are somewhere along the process of divorce - one is waiting for the paperwork to go through, the other will probably not get to that point for awhile. One doesn't seem to NEED support, doesn't talk about it and doesn't want to, so I support them by minding my own business. The other is more visibly stressed, more often stressed, and besides being willing to be open to talk if they need anything, I've let them know I will ask how they are every morning, and if they have anything they want to talk about, it's a good opportunity for them to be reminded they can have my ear/shoulder if needed.
You tell them you want to give them the time/space they need to process the big changes and loss they are going through, and you schedule dates as wanted. What works well for me, support them via IM, or coffee/lunch dates, or phone calls during the rest of the week, and try to concentrate on the two of YOU during your actual dates.
Note - I've also recently been supporting a partner through the death of a metamour - the same applies, things have gotten less painful for them over the last couple of months - I check in about how they are doing, they try to stay present during our dates, and don't always succeed. Try not to take things personally if you know things are due to grief. Try not to let your stress and worry about how they are doing affect your happiness to be with them.
Happiness will never come to those who fail to appreciate what they already have.