Causes for toilets to self-flush due to water pressure issues
Most houses have a 'stink pipe' - a 6&quot; pipe that discreetly goes up the side of the house and ends either under the eaves or pokes above the roof.  It's main job is to vent smells from the drains system.  However it also has a secondary job:
When you flush the toilet, the water rushing down the pipe can create a vacuum in the drains system if there is no pressure outlet to to maintain a pressure equilibrium in the system.  This is a second and most important purpose of the 'stink pipe' - it ensures the there is no build of pressure in the drains.  If this pipe becomes blocked or otherwise stops working however, you can find that flushing one toilet can affect the pressure inside the system and you will get funny things happening to other toilets / drains such as them emptying or overflowing as you've described.
First thing I would do is locate this pipe, and make sure nothing has happened to it - have you had any external works done on your home recently?
Secondly, I would locate where this pipe joins the system - there should be a drain cover, or at least a rodding point.  Remove this cover and flush the toilet as before.  If the downstairs toilet is not affected this time then it's highly likely that the stink pipe has been blocked somehow.
If this is the case, I would rod it from the top to ensure it's not blocked (birds nest, animal litter, leaves etc).  You'll need a good ladder and some patience.  If it's blocked, clear it and then get a filter cap for the top to ensure it doesn't happen again.
If that doesn't fix it, I would try rodding all my drains in both directions.  Make sure any blockages are cleared.  And if that doesn't work I'd get advice from a plumber on what to do!  Good luck.