As referenced in previous postings above, engines were captured and sent to the US, I believe mostly to Wright or Freeman Fields. I am not aware of any particular engines captured from crashed aircraft that were immediately sent. Being from Pittsburgh and having worked in the Homestead Works Steel mill in the late 1970s, I was surprised to come across a photograph of a Bf109 fuselage in that steel mill in 1944 as part of a War Bond tour. It was complete with a DB605 engine and it appeared to have North Africa markings. Alway wondered where that ended up. I do know that Smithsonian Air& Space (NASM) ended up with many of the engines over time as they became obsolete.
Many engines came as part of the aircraft and seemed to have been only examined as much to make the aircraft flyable, especially the recips. NASM currently has on display one Ju52 with 3 BMW132, one Bf109 with a DB605, an Fw190 with a BWM801, a Do335 with 2 DB603, an Ar234 with 2 Jumo-004, an Me262 with 2 Jumo-004, an Me162 with a Walter HKW, and an He219 with 2 DB603. We also have an Fw190D loaned to USAF Museum with a Jumo 213. In storage awaiting restoration are a Ta152 with a Jumo 213, an Ar196 with a BMW132, an Me410 with 2 DB603, a Ju388 with 2 BMW801 ( not sure which model), and an He162 with a BMW-003. I believe about a dozen extra Jumo-004 and/or BWM-003 were also brought over in 1945 to support testing the jet aircraft. I have seen in the engine storage area some BWM801 power eggs complete with cowlings.
I also recently helped prepare for loan to Japan a Mitsubishi Ha 42 Model 41 Ru Radial Engine. Very similar to a Wright 3350. There was little evidence that much was done with this engine, captured late in the war. It was to power a long range bomber under development.
One piece of distinct evidence of the US examining an engine accessory was the document created detailing the Kommandgerat from a BWM 801D2. It was issued in April 1945 by NACA, Aircraft Engine Research Laboratory. I was able to download a copy many years ago. Not sure if it is still on the Internet.
Tony