What you are describing is often from badly worn engagement dogs on the gears themselves. Seen more on the higher gears on street bikes that are continuously "power shifted". The corners of the dogs round off and no longer hold engagement.
I'm not sure how you'd get this on a dirt bike unless a previous owner was in the habit of ramming it into first and spinning the rear wheel to show off. Clutchless shifting between gears on a dirtbike is works OK as long as the throttle is eased off during shifts. With good oil and reasonable care a gearbox will last for years.
The only other damage I might expect is a bent shift fork. That makes for difficult shifting, but doesn't necessarily make the bike jump out of gear. Looks like you will need to split the cases to find out what's wrong.
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2011 EC250E
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