On the 8th of June or so, main colder air mass—if in places amounting to only more moderating "cooler air"—should begin to move and spread daily more southward, through until just after the summer solstice.
This with its being caused to move increasingly more directly south from the 8th through the 17th of June or so, with its at the same time gradually more slowed movement and pace more eastward.
No "Numerical Wx-forecast Models" have been consulted, where considering the generation of the general outlook included above as part of this post.