Adding an international co-investigator to a grant can look impressive, and increases the likelihood that the grant will have international impact.
However……some words of caution.
Set aside LOADS of time for liasing with the international university’s financial/research department on getting costs. As the grant body won’t be one that they are familiar with you may be asking for information that they are not use to calculating or providing (ever tried getting annual salary info from a US researcher? They don’t exist! They have term time and summer time wages that are not equivalent. They’ll also try and get you to cost for buying out time from a course. This is outrageously expensive and tell them to sod off!). Also, beware that some countries have much MUCH higher wages for researchers than the UK even if they are of an equivalent position to the PI which can skew the costs of the grants.
Secondly, quite frankly, they don’t really care. They have very little incentive to prioritise your requests because the grant isn’t coming to them (which is a problem with home nation CIs as well), and there is very little money in it for them as you’ll be requesting less staff time than the PI. I think this is the major factor in the slow and unenthusiastic response from international universities.
This info is particularly critical if you are aiming for a grant with a deadline. You’ll have zero experience of the other uni’s financial and research support staff and how prompt or enthusiastic they are. So, set aside 4-6 weeks for getting this info.