As I see it... Convection was and is being started by the ULL. Also at the same time, if during the summer months here in the Atlantic, sometimes when you get convection that is sustained over an area, it can form an area of low pressure in the lower levels.
CLOUDINESS AND SHOWERS OVER THE NORTHWESTERN CARIBBEAN SEA ARE
ASSOCIATED WITH A BROAD AND ELONGATED AREA OF LOW PRESSURE.
UPPER-LEVEL WINDS ARE CURRENTLY UNFAVORABLE FOR DEVELOPMENT...BUT
MAY BECOME MORE FAVORABLE DURING THE NEXT 48 HR AS THE SYSTEM
DRIFTS TOWARD THE NORTHWEST.
If there is no more convection associated with an area of low level pressure, then it will die off. The thunderstorms is pretty much the main part of what was and or is causing the low level pressure.
If the ULL is moving off to the East that doesn't mean that the the low level pressure will follow.
Lets look at the 00 hour of the 250mb wind/heights and the 500mb. These are two different levels of the atmosphere.
This first image we see the 200mb vorticity, we know the location of the upper level low and it shows up here on the this map.

Here is the 250 mb wind/heights 18Z GFS 00 Hour. We see currently that in the same area of the ull, winds are moving to the east and so is the ULL.

Here is the 500mb same run, same hour. There is not much wind in the area at all... Also we know that in high pressure rotates clockwise in the Northern Hemisphere. Looking at weak tropical systems in the tropics, it's usually the best to look at the mid to low level steering. In this case, our system is parked on the Western side of the High Pressure, with nothing to steer it out.

At the same time, if the thing that was sparking the convection, The ULL pulls out of the area and the low pressure isn't very strong, and there is no convergence at the surface, this area will die off and the the spin will die down.
We'll have to see how much longer convection is in the area and if it isn't then it won't.
But, please know this low level pressure is not going to shoot off to the north and east... Just the ULL, the thing that is sparking the convection is.