I've had this problem intermittently in the past, but it went away on its own. when I move the lever to shift to a smaller rear cog, nothing happens. there is no resistance. The lever moves freely, but nothing happens. If I keep at it, it eventually "catches" and I can shift pretty normally. There is a very slight amount of front-to-back play in the lever, and I find that if I push the lever slightly forward, it will catch and shift. It shifts into larger cogs with no problem. what could be going on. It is a dura ace shifter and it is about seven years old.

It may be that the cable is getting stuck somewhere between the shifter and the derailleur, so that the derailleur spring is not strong enough to pull the cable properly. You can shift ok the other direction because your hand is much stronger than the spring. There could be a kink/bend in the cable, rust, frayed cable ....
After 7 years, you are due for new cables & housing.

Homebrew is accurate - it can't be anything but a cable system problem. An entire replacement of the cable and housing is indeed a good idea, but if you have been unable to diagnose the problem yourself you may want to have a shop do the work to make sure the housing is cut, sized, terminated and seated properly, that the cable is lubricated if necessary and that the derailleur is properly adjusted.

If you want to do the work yourself I would suggest you start by familiarizing yourself with how the lever, cable system and derailleur work together. When you shift to larger cogs the lever pulls on the cable, which then moves the derailleur inward. When shifting outward to larger cogs the lever releases cable, which then has to be pulled through the housing by the derailleur using spring tension. If there is too much friction in the cable (or rarely if the spring is weak/broken) the derailleur cannot easily pull the cable through. Of course the system works the same for the front, but in most cases the derailleur moves outward when the lever pulls, inward when it releases.

It's fairly easy to isolate the problem. Shift to the largest rear cog and then, without pedaling, shift the lever to the smallest cog position. The cable will now be slack. Using your hands pull the cable back and forth on either side of a section of housing. If you have brifters (shift/brake levers) you will have to shift the lever back and forth while you hold onto the first section of cable. You should be able to detect where the friction is high. You may also see rust on the cable when it emerges from the housing. Friction may be caused by rust or dirt where cable passes through housing, kinked cable, distorted housing ends or cable stops, or broken cable strands. If the cable is rusted at all always replace both cable and housing. Study several tutorial/video on cable replacement (just Google replace rear derailleur cable) before you decide wheter to do the replacement yourself, and you will need good cable/housing cutters to do a neat, effective job.

Thank you so much for the explanation. I'm not with my bike right now, but i rode in the rain on Sunday. When i was cleaning my bike, I noticed there was some road gunk in a cable guide on the chainstay. When i get home, I'm going to take a toothbrush and clean that out and maybe put a drop of tri-flo in there to see if that helps.

In addition to the above suggestions a cleaning and re-lubrication of the derailleur itself is probably warranted. If you decide to replace the cables and housings (highly recommended), use good quality lined housings and die-drawn stainless cables. The last loop of cable and housing going to the derailleur frequently gets water and road grime in it so don't forget that one. For good performance it is essential that the housings are cut and fitted properly and that the correct ferrules are installed.

If you are going to ask for assistance it is adviseable to not translate complete instructions into guessing what the problem/solution is. Go through the cable system as I described, determine the problem, and then learn how to correctly resolve it. Randomly guessing what is wrong and how to fix it will only succeed in wasting a lot of time and possbibly overlooking something important.

Yes, sensei.

I have this problem each year on a bike I keep in the UK. The bike is not used for about 12 months, but I simply spray WD-40 into the lever mechanism and work it to and fro and this fixes the problem.

This is also what I recommend. But you probably do need new cables and cable housings. The housings collect dirt and corrosion and should be changed with new cables.
When shifting to smaller cogs the derailleur spring is pulling cable from the shifter. Any excess friction along the path of the cables can cause shift hesitation.

But try the WD40 trick first. If it helps you know you have excess friction in the shifter.

It sounds like this might be an STI-type lever.

When the OP "pushes the lever slightly foreward, it will catch and shift".
This tells me that the "foreward" movement might be causing the OP to move the lever a littl more slowly, which will usually allow a gummed pawl to catch the ratchet wheel.
These problems always show up with the arrival of colder weather, and the repair suggestions for this symptom are all over the map, from flooding with WD40 to immersion in hot oil.
I prefer to remove the hoods and immerse in hot oil, as this has never failed to give a very long second service life in my experience.
The still-hot shifter will need to be whirled at the end of rope or bungee to expell excess oil that would otherwise tend to leach out into the rubber lever hood.