My 2005 Dodge Caravan makes a popping noise near the right reaR wheel when put in reverse or drive

when was the last time you had the brakes checked most recommended checks would be suggested between 25,000 and 35,000 miles I generally have mine checked every summer to be safe if they havent been replaced since new they chances are you are down to the metal and that is the grinding noise you have to most like replaced the rear shoes and drums if sedan or if coupe replace rear disc and pads. from here in make sure you have them checked every summer to avoid replacing the drums and or discs these are expensive to replace where shoes and pads are minimal costs.. hope this helps

Does the noise happen just as the vehicle starts to move in either direction, or after just after you shift?

There is a small metal tab on the brake pad made to make noise
when the brakes are close to worn out,the noise goes away under
braking,I suspect the pads are in need of replacement.Make sure
the rear brakes get checked for proper adjustment at the same time.

Sounds like a broken flex plate.

REAR BRKS HANGING UP

It could well be in need of new brake pads however this would be noticable more when you apply the brakes if the sound is in the rear its not the front brakes it could also be your wheel hub bearings are past there used by date its a serious problem and needs attension right away

remove the road wheel and inspect your brake pads or shoes, possibly worn down to metal

The bearings are going bad in your colum

Depending on your mileage, if the transmission is mounted side ways (Front wheel Drive) the upper or top, or rear mount can wear out. This causes the engine and transmission to twist binding the linkage on the transmission, However I have not see a 2007 and if it was bad the drive train should still be under warranty if it's not over 100,000 miles. The tork on the other mounts could cause this noise you are speaking of. Also they use oil filled mounts to isolate the engine pulses so when the oil leaks out the motor mount goes metal to metal. However I think the brakes not holding tight so in theory a motor mount could cause this could cause it. A way to test it is put the transmission in forward gear then reverse if it makes a clunking sound then you have some wear somewhere. If you have another person put it in forward and reverse and watch the engine you can find where the noise is coming from. However I will see if it is listed as an after market part and give you a link that you can look at and the price for each. After looking at one of the mounts this may not be your problem here is the link and the price for one of the mounts.
http://shop.advanceautoparts.com/webapp/wcs/stores/servlet/product_Engine-Mount-Anchor_3264483-P_27_R%7CGRPENGPAMS____

does the van shake or vibrate? does the van have tires on it ?start by switching the left rear with the right rear test drive and listen for soun if no sound then the tire has a bad steel belt replace tire if the sound is still present. jack up van and spin tire if no sound is heard replace wheel bearing and if there is still a sound look into brake drum for possible springs or clips that may have fallen into drum causing noise but i would say its more than likely the rear wheel bearing to change that u must buy a new bearing and follow instructions on replacing good