According to Newton's Second Law, if you keep applying a force you should continue to produce an acceleration. However, we observe that sea breezes tend to attain a fairly steady strength by noontime or so and fail to continue to intensify. An additional force or forces develop to oppose the pressure gradient force. Answer will appear here. The vertical motions, meanwhile, also affect temperatures.
Downward motion causes air to warm as it moves toward levels with higher pressure.
Usually, this causes a net decrease in density in the air column, and consequently a decrease in pressure at the base of the column. The opposite situation takes place over land, where the rising motion cools the air and tends to cause higher pressure near the surface. So the winds themselves alter the temperature distribution, and thereby affect the pressure gradients, reducing the driving force of the circulation. The other forces, friction and coriolis, also come into play as the wind begins to blow. Of these two, friction is more important for the sea breeze.