I have to say having skimmed briefly over the report you posted a link to I'm still sceptical about its absolute safety although it does lay some serious concerns to rest.
At the bottom of page ES5 it says: "Actors with the highest exposure to mineral oil had a statistically significant decrease in one pulmonary function parameter - forced vital capacity".
I believe FVC is a measure of the total capacity of the lungs. Although this doesn't directly suggest it causes asthma, it does suggest that the ability of the lungs to work under stressful situations eg. running, may be impaired as there is a reduction in the amount of air that can be inhaled / exhaled from the lungs as a result of high exposure.
However, you are right it appears that provided medical grade mineral oil is used, the carcinogenic compounds commonly found in oil of the type used for lubrication have been removed.