I'm really new at pen making, but fountain pens are what I am most interested in and have done the most challenging work on so far.
Since I'm new, take what I'm saying with a grain of salt as it may only apply to the ones I've worked on and might not work at all for you on yours.
Sometimes they write perfectly as the come with the kit and sometimes I have to put a lot of effort into making them work the way I want them to (IMO a fountain pen should easily and immediately write under nothing but it's own weight).
Most of the kits have pretty cheap nib/feed assemblies in them (which makes sense since you wouldn't expect a 50 dollar nib in a 10 or15 dollar kit). Two problems I find here, probably quality control related since they are not consistent from one to another, are in the nib itself having out of line points or too narrow or plugged a slit in it and the feed itself not being right to fit against the nib and feed ink smoothly or having a manufacturing defect in it.
Before you do anything wash the whole thing thoroughly in lukewarm water with a little dish wash detergent in it, and rinse it with clear water. You can just soak it and use a converter to pump the water through or you can fully dissemble it and wash the parts with a soft toothbrush. A lot of problems can be from having manufacturing gunk left in the nib and feed.
Carefully inspect the nib itself with a magnifying glass and a light behind it to see if there is anything in the slit that could interfere with the ink flow and look at it head on from the pointy end to see if the two tines are in perfect alignment with each other (if you find a problem here there are instructions on how to correct it available all over the internet, but be rather careful your first time).
If you're satisfied with the nib itself reassemble everything, install your cartridge or fill your converter, hold it pointed down for a minute then try to write with it, if it doesn't prime itself prime the nib and feed by dipping it in an ink bottle or forcing just a little out of the cartridge or converter till you see it wetting the nib. You may drip some out so keep your work surface appropriately covered while doing this.
It should write now, and should write under its own weight if the nib is fully corrected.
Now store it upright (with the cap on) and see if it will start and write at intervals of about 1, 2, 8, and 24 hours. If it does you've solved the problem.
If it doesn't, let it dry again (stops starting) and slip a piece of regular typewriter paper just a small ways under the nib (maybe to the back of the front tapered part) and see if that starts it. If it does you probably have a problem with the nib to feed fit or the feed itself.
To check this let it go dry, then without trying to write more with it pull the nib and feed out together without letting them change position relative to each other and carefully separate them and look inside the nib to see where the ink is still wet and where it is dry or just not there. This is the point whre the problem lies, examine the feed at that point to see if there is any visible problem that would cause the ink to slow or stop there. If something is obvious then you can figure out how to solve it (but beware, it is far easier to do too much at a time and ruin the thing trying to do it in one step than you might think. Do whatever you do just a tiny bit at a time). There are instructions all over the web about how to correct this, rad all of them you can find and apply as you think needed. (doesn't hurt to buy a few cheap nib/section assemblies to experiment on).
This is, of course, from my very limited experience and may not address your problem at all. It's just what has worked for me so far. I've heard inks can make a big difference as well, but I've had no experience with most of them.
