You will have no problems .
Since I travel a LOT , I need to renew my passport every year almost . I have some long term visas ( Including 10 year visa ) . I just carry the ones I need together .
Further Information regarding valid visa in expired passports :
Some countries require that you will renew the visa if the passport is changed . Most don't . ( I never actually encountered it in case you have the old one )
That is the case for EXPIRED passport - not for LOST or DAMAGED .There is a difference.
When you renew your passport , the consulate ( or respective minister ) is obliged by international law to give you back the old one providing it has valid visas from 3rd countries . legally speaking , the visas in your passport are properties of the country that issued them, and no other entity can invalidate or destroy them .
The most common way of invalidating a passport is to cut the "ears" or corners of external cover . That way no visa is damaged .
If you do not have valid visas some entities will punch a whole through the passport and give it back if specifically ASKED to.
In most cases ( depending on country of course ) , when applying for a new passport - the old passport will be validated with some kind of a note such as
" this passport is replacing an Expired document NO. XXX " of one kind or another . This ties both old and new passport together .
Some countries require further documents on arrival, such as Australia and the Airline is actually the one to prepare it .
from the Australian Consulate immigration Info page
If you have a valid visa associated with an expired passport (i.e.you
have been issued with a new passport since your current valid visa was
granted), you may travel carrying your old and new passports together
and the airline will arrange to have the details updated at time of
check-in. If so, you should check-in a little earlier to avoid delays.
So again - the direct answer - You will have no trouble as long as the visa page on old passport is valid and not physically damaged , and you have that with you on arrival .