How important is a climate controlled vehicle for transporting oil and acrylic paintings?
As long as paintings do not experience sharp fluctuations in temperature and RH in going to and from an exhibition/storage space to the truck/van/etc. AND as long as one remains relatively close to the ideal temp and RH range you should be alright. Actually it is often during transportation that these fluctuations can occur most readily unless the paintings are covered with polyethylene sheeting and properly crated. There are some good references listed in our bibliography at the bottom of this document which you can find in our resources section: https://www.artcons.udel.edu/mitra/Documents/MITRA_Storage_Exh_Handling.pdf
The answer is obviously very situational but generally, it is important. Transporting paintings in the Spring or Fall and in a moderate temp and RH, may not be a huge issue. Transporting them during the Winter, it is vastly important as the glass transition temp (think the brittleness of painting under the current condition) massively increases at low temps. This is true of both oil and acrylic paintings. Transporting them the hot Summer is also a recipe for disaster. Even appropriate wrapping may not alleviate potential issues. We have seen acrylic paintings and oil paintings with acrylic varnishes with global impressed damage to the surface because the temp was high enough that wrapping materials deformed and stuck to the surface. This is irreparable.
Additionally wide fulgurations in temp and RH are most likely when moving the paintings from a stable environment to one that is both uncontrolled and likely more extreme. In short, it is always best to transport any paintings that are important to you any way in a climate controlled vehicle, and it is essential in many situations.