These questions never end somehow !
Sometimes I get slow rooting cuttings. Some roots appear in the baggie and I wait for more roots before moving it to Perlite-Vermiculite in a cup. While waiting for more roots, the previously appeared few roots start browning and becoming thin. What to do in case like this? And, why do the roots start browning in the bag, lack of moisture in baggie or what?

What I do is as soon as I have roots (even one) that are 1/8 - 1/4 inch in length, I move them to Verm/Perl or other rooting media. If you wait too long they will brown like you observed and eventually die. The cause is either - too much moisture (rot) or not enough (drying out). I've seen both. I don't find any additional benefit from letting more roots develop before putting them in cups - in fact, you risk the problems you are seeing (this was another one of those lessons I learned the hard way). Once the roots die, they will not regenerate. It will be like starting the rooting process all over.

SteveNJ
In the beginning I used to that, not because of this knowledge but because I was impatient. So impatient then became unintended virtue. Now I will do it based on this knowledge and move to cup and see how it works. So, now the only issue left is the Murphy's law...
Thanks.