Find next bigger number with same combination of digits
You convert number to array can be simplified by first converting the number to a string and then splitting on nothing:

var convertToNumArray = function(num){
    return num.toString().split('')
}


This will of course convert it to an array of strings, but that will work perfectly fine for your purposes (or you can always throw in a map to convert back to ints).

Your first loop (noted with this comment):

// iterate from right most number to find out the 
// number which is less than previous one.


You don't need the else { continue }.  It is at the end of the loop, so continuing is guaranteed.  Sometimes explicit is nice but I would say not to the extend of writing code that doesn't actually do anything: that just makes for more code to read through.

Also, I don't think you have accounted for the possibility of a number that doesn't have a next value.  For instance, how does your code handle 54321?  Your swap index will stay at the initialized value of zero.  Either way, I think the smart thing to do is to detect this event happening and exit prematurely, as you already have the answer: there is no room for changes.

Your second for loop looks solid.  I might organize it a little differently but no actual complaints: just a different coding style.

For your last bit, I wouldn't do the sort yourself.  Use the built in javascript sort.  You can slice off the end of the array, use sort() (keep in mind that it sorts asciibetically, but that shouldn't hurt you here), and then splice it back onto the end.