I have been in a similar situation. I spent quite a long time comparing different companies and after my calculations I decided to use
Movers Manhattan I didn't expect their team to be so helpful, I overslept that day and their guys helped me pack my stuff, they also packed and moved everything carefully. Thanks to them the whole process was surprisingly smooth and stress free during an emotionally difficult time. Having movers that I could trust allowed me to focus on other issues that came up that day rather than the logistics. I am confident that Manhattan Movers will do the same for your relatives move. I wish you all the best during this move.