I installed an extended thumb safety on my Springfield mil-spec by fitting and testing it properly with the sear engagement.  Anyway, it works properly and safely and passes all the safety tests.  Now what I would like to know is how to make it move up and down between the safe and fire positions more smoothly, i.e. it is kind of difficult to move back and forth compared to the other thumb safeties on my custom-built 1911's.

You need to check the detent (read may need to be polished if groved), the safety detent hole for a ridge, and make sure that the safety is correctly tuned to the frame/FCG.

Also, a little grease may held the safety mate to the frame.  And, if this is a ambi, then don't forget to tune the weak side transfer pad, and apply a little grease between it and the frame (under the grip).

Quoted: You need to check the detent (read may need to be polished if groved), the safety detent hole for a ridge, and make sure that the safety is correctly tuned to the frame/FCG. Also, a little grease may held the safety mate to the frame.  And, if this is a ambi, then don't forget to tune the weak side transfer pad, and apply a little grease between it and the frame (under the grip).

I checked the detent, and it seems fine, so I don't want to risk taking any material off that area.  How can that area be polished, though?

The safety works fine, i.e. it moves up and down, and keeps the pistol from firing when up/on/safe and allows the pistol to fire when down/off/fire.  What more do I need to check to "correctly tune it to the frame/FCG"?

Already some good suggestions given to you, but sometimes all you need to do is work it to mate it in. Use a good oil and watch TV after CLEARING the weapon and work it on and off. I have had some excellent results just doing that. Remember tho it should not snick off to easy, especially on a carry gun, you will not be at the range. I like a nice very "snicky" safety on my carry pieces personally and usually stay away from the overly extended ones, I like a nice deep checkering more myself.

Quoted: Already some good suggestions given to you, but sometimes all you need to do is work it to mate it in. Use a good oil and watch TV after CLEARING the weapon and work it on and off. I have had some excellent results just doing that.

That is actually doing the trick, the more I flick it on and off, the smoother it gets now. Thanks.

Actually, you want a nice, crisp click between on and off and vice versa.  NOT SUPER HEAVY, but very positive.

It is positive between ON and OFF, I just wanted smoother travel in between those two positions.