I'm not saying I wasn't wrong... initially. I misread "transition into" as a fade/dissolve. I have since reread the post more carefully.
Of course you're right, as far as this: you can add a compound or secondary storyline and that will create the effect from the transition the OP has asked about (as will applying any of the Stencil/Silhouette blend modes for masking techniques, which will cut through everything underneath the "layer" with the compositing blend setting.) Grouping (aka, compound clip) will stop those related actions, isolating them to the contents of the group.
The necessity of grouping is due to the fact that Transitions and Titles copy the storyline (or what's "visible" below their level) into the template (within the scope of the group they exist in... i.e., storyline, compound clip, secondary storyline -- there's actually a placeholder provided in Motion -- these allow for direct interactions with created material and the original media within the template). This type of thing would be better created as an Effect (which uses only the clip's source media [another placeholder.]) And so, I did (see below.)
However, working the (default) Flash transition this way (compounding) does not eliminate the whole screen blanking to white for that frame or three since the effect is affected by a solid white canvas covering rectangle regardless of source (which is very likely the effect most people want most of the time, but not the OP, in this case.)
I was killing time (watching "The Horror of Dracula") when I put this together -- everyone is welcome to give it a try:
http://sight-creations.com/fxexchange/Fade-Flash-Fade.zip (not a brilliant name -- and -- it doesn't do a bloom -- but -- it's free). This needs to be installed as a Motion Templates Effect. Parameters include length of fade (0 to about 70 frames even though the "scale" is 0 - 100) and the amount of blur for the "flash in/flash out" effect (a gaussian blur from about 64 min to 600 max; even though the scale is 10-100). "Everything" is relative to the length of fade setting. This effect only whites out and blurs whatever the source of the clip is and doesn't attempt to flash the whole screen (initial luma values have an impact on full screen media). I recommend shorter fade times with shorter clips (designed for 150 frames with max length fades of 70 frames either side of the 10 frame "neutral zone" (which is the segment that will get time stretched when length is changed). It's effective with Titles and Generators as well and best used on connected clips, not the main storyline (try it with connected clip Placeholder generator 🙂 -- "kinda" beam-me-down-scotty going in and nuclear annihilation on the way out... LOL ).
If you would be interested in altering the default Flash transition template in this fashion, in Motion, delete the group containing the rectangles and to the group containing the Transition A and B sources, add a Hue Saturation filter and keyframe the Value parameter to it's maximum value at the edge of the source changes and back to default at the end and Save (it will automatically be created as "Flash copy" -- you cannot accidentally overwrite the default templates).