If the situation permits, I don't see why not.
Let's put these through a couple scenarios first.
A: You and the enemy Mid duke it out while the Jungle is at Red. You die on your side of the lane, but the wave is still in the middle. The enemy mid is low, but shoves quick and walks away. Your jungler can a) cancel doing red buff, meaning he gains nothing for his hp lost, and try to intercept the enemy mid laner. If the jungler has no gap closes, the likeliness of getting the kill is low, near impossible if they still have flash. The jungler loses out on gold and xp and the only real option that doesn't give a negative is evening the waves. or b) finish red buff and then either even the wave or continue farming. situation "b" is the safest for the jungler and contains far less risk.
B: enemy mid dives you and wins and ends up low. Jungler is at red. This situation presents the most opportunity for securing the kill, even without a gap close. Likely the enemy laner is shoved in and may get greedy for another wave as they usually flip back and push into their tower from this point.
or C: You dive and fail. Under very rare circumstances can and will a jungler attempt to secure a kill. This almost requires either range or a gap closer and flash to prevent dying under tower. This also very risky as it can end with the enemy mid gaining a double kill, setting you behind even more.
All these situations assume that the jungler is with "buff range", meaning halfway between lanes. If the jungler is farther out, say at gromp or krugs, or even ganking another lane, it is still not his responsibility.
With that out of the way, securing the kill doesn't prevent the mid from feeding. The enemy mid has still gained a gold advantage, and any kill gold gained goes towards the jungle, meaning the mid has gained nothing.
There's also another thing. The jungler is not a baby sitter. He is not there as a get out of jail free card. The wisest use of a junglers time is doing these 3 things: 1) Counterjungling/ganking when the enemy jungler appears. Essentially, keeping track of the jungler, much like laners track their opponents. This can be way more difficult.
2) ganking winning lanes, especially ones where the opponent insist on pushing. This insures that the winning laners keep a solid lead, making them stronger for the mid and late game.
3) Farming. Especially If all lanes are a) late game champs b) shoving in when the enemy jungler is farm heavy (yi, fiddle, jax, rengar) and applying pressure against the jungler if possible. (after 6 against these champs the jungler must anticipate strong ganks)
Hopefully this clears up what junglers will do.
