I had lots of trouble with this when my GZ250 was new. Here is my take.
It's not broken. Lots of small bikes have this problem. I have ridden Hondas that did exactly the ssme thing.
The GZ250 owners manual says you need to be in first gear when the GZ250 goes below 9 mph. Some experimentation show this to be somewhat variable but a pretty good general rule of thumb. The slower you go below this speed the more likely you are to miss the down shift and wind up in second, third or neutral. You can even get caught part way in to first and have the bike jump to neutral when start moving.
So how do you deal with it happens to you? Don't force it. You could break something and then you get to push your bike. You can move the bike forward or back a little and take some of the pressure off the gears. This works pretty good on level ground like in your garage I do this just about every morning to get it to shit into neutral for a warm up. The primary method I use in traffic is to give it a little clutch just into the first part of the friction zone while applying some pressure to the shifter. This will get things moving in the transmission and when it lines up in the right place you can down shift. Repeat until you get into first.
When you stop make it a habit to check to see if you are in first gear. After you have put as many miles on your GZ250 as I have you will notice that the shifter feels different when your in first gear than it does when your in any other gear. It feels harder. It also makes a little clunk sound going into first.
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