Designing PSU: PSU Curent Limit Latching issue
There are multiple troublesome elements in the design, it's hard to know where to start. There is no latch up in the circuit you provided, but when associated with your external power supply circuit (https://www.youtube.com/watch?v=Z_YtQgTnvkU), it is a poor overall design.

I am assuming your design intent here is to provide a CV/CC design, where the voltage is regulated until you reach the current limit, and then the current is regulated. Your design does not do this.
In your design, you attempt to drop the voltage to zero by shorting out the 'reference' output with Q1. But the time delay in shorting out the I(set) pin defeats most of you objectives.  


The LT3080 needs to be included within your feedback path. It currently is not since you are trying to set I(set) and with long delays when you have a 22 uF capacitor connected to it. 
When you reach your current limit you move completely out of any regulation feedback. This might work if the only overcurrent that ever occurs is a resistive short circuit. What about modeling your CV/CC approach when you have a string of LEDs attached and you want to regulate the current through them. Any decent CV/CC power supply can do this.  
Your current sensing is via an RC delay, which as noted in the other answer almost guarantees you will oscillate between CV and no CV. 


Suggestions:


Feed your reference voltage V-SET via a 1K Ohm series resistor to IC3A.
Connect you current limit detection, Q1, to IC3A(+). This will now reduce the voltage as the current limit is reached. To get to zero volts you would need to change Q1 to an N-Channel FET of course, or have a negative supply for Q1. 
Get rid of the 22 uF on I(set)
Connect the feedback resistor R42/43 to the actual output of the LT3080's to include them in the feedback loop.  
While it was a good thought to use an LM334 to supply the minimum current for the LT3080's, it does not work down to zero volts. If you use this then setting voltages below 1 V won't work, and this is probably the reason for the non-linearity you see at low currents/voltages. If you expect the power supply CV/CC support to work down to mV and mA levels then you need to move your current detection to the outputs of the LT3080's and draw the minimum LT3080 required current before the detection point. A simple N-Channel Enhancement mode FET would do the job, though you need a negative supply to get regulation down to zero volts.