One idea put forward was to conduct PCRs in the field. PCRs are polymerase chain reactions, a technique used to multiply copies of DNA sequences. The DNA can be collected from nearly anything, but for noninvasive primate studies, it's typically collected from fecal samples. Field PCR kits are readily available.
There are international standards restricting the trade and transport of tissue of exotic and endangered species for fear it will encourage further still the slaughter of living examples. PCRs generate synthetic DNA that is not included in this ban.
Some host countries are wary of allowing DNA to be exported for fear of not sharing in the intellectual and financial gain derived from the study of this material, a form of scientific colonialism. Conducting the PCR in the host country rather than exporting the original DNA can allay this problem, if local scientists are trained and employed in the lab work.
The PCR is usually more robust the fresher the sample.
Importing synthetic DNA is less problematic for the US Dept of Agriculture than fecal and saliva samples.
Ice and electricity are required to conduct the PCRs. While certainly usually available in the host countries, these are typically in short supply in the field. Some teams have brought in generators and liquid nitrogen, but transport of fecal and saliva samples is not any more difficult than this, and probably less so.