Here's one thing you can try... It won't be 100% straight-forward but then again it might be useful...
Logic lets you have "time signature alternatives". So let's say you have four tracks of constant time signature throughout. Using your example of...
...set Time Sig Alt 1 to be 3/2. Compose a part of your first track. Then change to Alt 2, set it to 4/2 and compose a section of the next instrument. Rinse, lather, repeat for the additional tracks. And as you need to review/edit/add to each part, change to the appropriate Time Sig alternative for that track.
By switching to the different Alts you will be able to print out individual parts, each with their own sig. But as has already been mentioned, you won't be able to print out the score and have it reflect different sigs for each staff.
As much as I love Logic's score editor, this isn't the kind of thing it's set up to do; part of the reason for this is that in Logic, bar lines are 'global'. And even though you can trick Logic into printing out different time signatures on each track and turn off the display of bar lines entirely, any custom-entered bar lines placed on a staff will appear on all others. So that makes it a moot point to learn those tricks.
So to write the piece in Logic you'll have to deal with the chore of switching time sigs and then move to some other notation program to print out your score.
(BTW, hello Rounik!)