I've been trying to propagate some cuttings. Rooted them and then kept them indoors under constant light and clear cover. Aired and misted lightly daily. They seemed to be doing well. A few people advised to try to start transitioning them outside by taking out in partly shaded area for few hours. I decided to take out four, two seemed okay but two started to wilt. I've brought them all back inside and put them back into makeshift greenhouse. Any advice? The first two are before and after placing them outside yesterday. The last two photos are of the ones that started to wilt, today. One seems to be okay, a few leaves a bit curled but the other one is still looking pretty sad.

I've had this happen to me and it has nothing to do with what you did. In fact I've been rooting some cuttings outside on my porch. Some are doing ok and some are not.
Some cuttings lose their vitality before they had a chance to root.
I like to use deeper containers to root my cuttings, especially if it gets too hot outside and the last thing you want to do is overwater.
I would say keep the soil moist at around 70 degrees, transfer when ready to a deeper container before taking them outside.

IMO They look too young to go directly outside.  
When changing a young plant's environment you need to do it in steps and let it get used to the transition and get them to harden first. 
Start by opening the cover so it gets used to room R.H.  You may still have to mist it or close the cover if the leaves start to wilt.
Some plants grow strong and transition fast some really need to be babied.
 

Thank you Sas and pino. I have 9 cuttings I am trying to propagate. I took 4 of them out in shaded area, 2 seemed okay but I'm also thinking that I took them out too early as pino is suggesting. I'll first try leaving top open as suggested and then start putting in window with indirect sunlight before taking them outside? So far they've been under artificial light. Is there anything I can do to save the one with leaves withering aside from putting back in covered box? Is it okay if they fall off?

There is still a chance that it might recover, but might take time. All you need is one green shoot to push out new leaves. If the humidity and temp are right.
I wouldn't disturb it nor give up on it until the summer is over. Good Luck with it.
I had one cutting for the past year. It had a green bud, but was unable to push new growth. Last summer it formed only two leaves.
This season it looks like it might take off.

Thanks Sas, good luck to you too! This is my second time trying, first time I tried I lost the cuttings. I had posted about it earlier, had taken these from father's trees. My parents had since moved and my father passed away not too long ago so badly want at least a few to succeed this time around.

Sorry about your dad.
My rate of success might have been two or three out of four and in some cases one out of four. In the end it depends on the quality and freshness of the cuttings.
If I really wanted to make sure I get good results, I would use a self watering pot the type sold at Walmart, fill it up with some moisture control potting soil from Home Depot. I would then insert the cutting in the soil and water from below. In order to insure enough humidity, I would cover with a plastic cup and put the pot near a sunny window.
I would check the reservoir to make sure it always have enough water and just leave alone. Some varieties might take up to 90 days before they show results.
I would choose about a one gallon pot in order not to have to repot the same season. For me, repotting newly rooted figs is always risky as the plant could get shocked.
I'm rooting a whole bunch of cuttings this way on my porch today, and despite that being treated equal, not all of them will make it.

Thanks Sas, I'll look into those self watering pots and maybe try that with at least a few. I think I need to start putting them near a sunny window and then slowly start leaving them uncovered more and more. I'll post with an update, probably also be asking for more advice.

This one has been sitting on my floor since before Christmas. I will be taking it out soon.

Had that happen to one of my rooted cuttings and it's possible for it to recover.  When I got home from work and found the leaves wilting, I took the plant inside the garage and cover it up with a plastic bag.  All the leaves dried up and dropped but the plant eventually recovered after about a week.  That was 1.5 years ago and now it's in my backyard standing about 7 feet tall with lots of figs on it.

Here some rooting on my porch. They sat in my garage for about two months in plastic bags while I had a broken foot. Thought that I lost them all and decided to put each variety in one pot.
This is the result after three weeks. I will probably separate them after they go dormant.

Thanks kevint8 and Sas! Gives me some hope for these. Hopefully the one with withered leaves will recover. I'll keep it covered and see how it does. A couple questions I have regarding leaves curling and brown edges. I've been careful not to over or underwater. What would a slight curling indicate? These are plants which are in a covered, clear box. A few of the leaves seem to be showing a bit of curling on edges. A few have browning on edges. First photo is all the cuttings. Second is example of curling, kinda difficult to make out in photo. Third is of one of leaves with brown edges. Not sure if it is all due to sun exposure. I did keep box next to sunny window for a couple hours on two consecutive days a few days back and had put a few of the plants outside as I mentioned earlier.

As long as you have secondary active growth, I would not worry about the shape of these leaves. The cuttings are too young to have perfect leaves. Keep them in shade/ partial sun for an extended period until you see strong growth and then move them to a permanent location. If in full sun, I would not put them in anything smaller than a 5 gallon pot.
Remember that the Texas sun is brutal and you could overheat the roots in small containers.

Thanks for all your advice Sas. I have been concerned about the Texas heat too. I know once they've been established they can do very well in our area but yes, the heat may be too much for these plants at this stage. I'm going to first try to slowly transition them out of their "greenhouse" indoors. I've been told several times to try Pro-Mix HP soil, so will try that out and with larger pots. I have an indoor sunroom so I may try leaving them in there a while once they seem to be showing more vigorous growth and do okay without being covered. At that point I'll start trying to transition outdoors.