Typical causes of the bridging you describe is excessive top side reflow temperatures. I would suggest you reduce you top-side temperature and increase the subzone temperature. This will reduce the thermal stresses on the component and reduce the chances of the component warping.

My initial thoughts are that you are getting warpage or possible popcorning of the packages. Have moisture sensitivity protocols been followed for the BGAs?

Take a methodical approach. First, look at the board bond pads. Board shops often (too often) make changes in the design and do not communicate this to the end customer. Ensure that the spacing between bond pads is uniform throughout under the BGA

Next, take a look at the BGA. Is the spacing between the balls uniform? Deliberate changes (such as an array that is not fully populated) or unintended changes could drive this behavior.

Finally, do the same thing with your stencil pattern.

If everything looks good, then I would start to suspect out of plane deformation of the BGA (aka, potato chipping) during the reflow process.

Sure, blame it on the reflow oven and profile - because that's where the bridges happened, right? In reality, you might want to check out the solder paste.

First of all, make sure that the profile you are using conforms to what the solder paste manufacturer specifies - you may be suffering from "hot slump".

Also, be sure you have the correct solder paste deposit height - yours might be excessive. Anyone assembling with area arrays should have 3-D post-print inspection, at least on an AQL basis. If these actions fail, it might be time to evaluate a new solder paste.