Assuming you are with Express...
You can put in for a hardship transfer, but all that does is move you to the "top of the list" when an opening occurs at the station you are requesting transfer to. If there is no opening, you wait. I knew two Couriers who had different circumstances and applied for hardship transfers - in both cases they took almost a year to get a position in the station they wanted.
If you are full-time, Express HR will tell you to take FMLA time (unpaid) and take care of your family issue - your job will be waiting for you at your CURRENT station.
If you are part-time, federal FMLA doesn't apply (you need to have worked 1250 hours in the past 12 months), you'd be left taking Express medical leave which I believe is limited to 90 days. Beyond that, and you are out of a job.
Talk to Express HR and give them the full story, they'll let you know what your options are.
If HR gives you a thumbs down, your next step would be to go to the media, and give them a "sob story". FedEx hates negative publicity, and having a story pop up in the media that "FedEx doesn't care", may just be enough to motivate them to bend the rules and get you to where you want to be.
First - Express HR
Second (if negative response), news media... and cry a lot during the on camera interview.