What is the easiest way to install 4" recessed lights in 6" holes?
Four inch and six inch recessed fixtures consist of two main parts - the can and the trim. They need to match.

The first issue will be getting the old can out. If it is old-work style, it may be held in just by pressure clips on the sides, fairly easy to remove. If it is new-work style, it will be attached to framing members, either directly or with a brace system. More complicated to remove and usually requiring cutting a bigger hole to get to it.

Even if it is old-work, a 6" hole differs from a 4" hole by 1" on each side. Cutting a 1" ring from drywall is hard, leaves you with a fragile piece, and will be difficult to tape and mud smoothly, and hard to brace properly.

I would suggest cutting a somewhat larger square out of the existing ceiling area. This will allow you to remove a new work can, if that is what you have. Leave at least 2" on each side of the hole. I would cut the new 4" hole (check the fixture for exact hole size) before mounting the patch to the ceiling.

Once you have a section out of the ceiling, you may be able to reach the framing to attach a new-work type unit. More likely, you will use an old-work style.

In either case, you will need to brace the new patch to provide strength to hold up the patch and possibly the new fixture. You can do this by using furring strips that are at least 6" longer than the size of your patch. The furring strips will extend on either side of the patch. 

The strips should be screwed to the ceiling beyond the patch and the patch should then be screwed to the strips. I would use 4 strips, all parallel, two on the outside edge of the patch and two just a bit wider than the location of the new 4" hole.

Once the patch is up, tape and mud (spackle) the edges.

Finally, you might want to think about moving the new fixtures to a slightly different spot to get a solid surface to attach to. This may require some adjustment in wiring.