First, as Aloysius says above, you should stay away from the ends of the mortise during bulk removal.
1/16 of an inch or 1-2mm should be sufficient.
Now, when you have done a bunch of chopping, and you have chips clogging the mortise, you need to clear them out, as you said.
Use a bench chisel that is one "size" below your mortising chisel.
For example, a 1/4 inch if you are doing a 3/8 inch mortise.
Guide the chisel with the thumb and forefinger of your non-dominant hand as you work it into the chips.
Then, still lightly pinching the chisel, rest your non-dominant forefinger on the board and use it as fulcrum as you lever the waste out of the hole.
The chisel should not touch the end of the mortise if you do this correctly.
woodworking.stackexchange.com/users/600/Mr. Kevin
Don't lever.
Pare straight downward into the opening, taking it in multiple passes if necessary, and letting the chisel act as a wedge to do the work.
woodworking.stackexchange.com/users/91/keshlam
Paul Sellers is a master craftsman, author of several books and some very good video tutorials.
He does several on cutting mortises - this one probably being most relevant to this topic.
https://www.youtube.com/watch?v=LPBkO2chZxk
I've learned a lot from watching his videos, still dont have the skill to implement the lessons very well as yet - very much a novice at this.
woodworking.stackexchange.com/users/1637/FreudianSlip
Are you looking to use only hand tools?
If you have a router, this can get the majority of the mortise bottom flat with a flat bottom bit.
Then the corners will be easier to chisel out.
A drill with a forstner bit can be used to accomplish this as well.
If you have a drill press, that would be ideal.
There are also hollow chisel mortising attachments you can buy for drill presses that will make perfectly square mortises, and many woodoworkers will "drag" this along the bottom to achieve a relatively flat mortise bottom.
If you want to use only hand tools, a router plane would be capable of achieving this, coming from both directions to get into the corners.
woodworking.stackexchange.com/users/2753/Jacob Edmond
Borrowing from my own answer about this very topic : Auger Bits Whether you are going for traditional techniques or powered/hand hybrid auger bits would be a tool of choice for this.
Using auger bits that are the exact width of you hole you are trying to make, in a brace and bit or drill, you can easily remove waste and not have to worry about going "outside the lines".
Stagger the holes and you will get a similar shape like that of the ones you have pictured. From Chris' Project Page As you can see from the above image (mortise in a workbench)
most of the wood was removed with auger bit.
The screw tip and the shape help to naturally create straight holes.
I'm really looking for a technique which allows me to reliably clear the debris without risking damage Be careful when/if you are exiting out the other side.
To mitigate damage you can Used waste/scrap wood (clamp well) Count
the rotations it takes until the screw (eye) just shows proud on the other side (A Roy Underhill suggestion).
Mark
the depth of your bits using masking tape.
Feel for the screw/eye coming out
Most of those suggestions will work in tandem.
This way you can turn the work over and have cleaner edges on the outside.
woodworking.stackexchange.com/users/128/Matt
I've watched a couple of videos of making mortises.     What technique should I use to remove (bulk) waste from the hole without damaging the ends of the mortise hole?   Clearly levering against them will bruise or dent the wood.  I'm really looking for a technique which allows me to reliably clear the debris without risking damage.   I'm especially looking for answers more specific than "be careful".  Edit: I'm not talking about the paring stage, I'm talking about bulk waste removal.
How to avoid damaging the ends of mortise holes when chiseling?
woodworking.stackexchange.com
<chisel><mortise-and-tenon>
woodworking.stackexchange.com/questions/4723
woodworking.stackexchange.com/users/2557/James Youngman
to avoid damaging the ends of mortise holes when chiseling you can use a bench chisel one size below the mortising chisel. A router will certainly be a big help as would a forstner bit. There are also hollow chisel mortising attachments fro drill presses. If using only hand tools, a router plane would be able to do it. When using chisels, stay away from the edges during bulk removal in a mortice. Use a bench chisel one size below your mortising chisel. Don't lever. Pare straight downward into the opening, taking it in multiple passes if necessary, and letting the chisel act as a wedge to do the work. A router can get the majority of the mortise bottom flat with a flat bottom bit.  A drill with a forstner bit can be used to accomplish this as well.  A drill press would be ideal. Hollow chisel mortising attachments  for drill presses make perfectly square mortises, and many woodworkers "drag" this along the bottom to achieve a relatively flat mortise bottom.  If you want to use only hand tools, a router plane would be capable of achieving this, coming from both directions to get into the corners.
Stay away from the edges during bulk removal in a mortice. Use a bench chisel one size below your mortising chisel. Don't lever. Pare straight downward into the opening, taking it in multiple passes if necessary, and letting the chisel act as a wedge to do the work.  A router can get the majority of the mortise bottom flat with a flat bottom bit.  A drill with a forstner bit can be used to accomplish this as well.  A drill press would be ideal. Hollow chisel mortising attachments  for drill presses make perfectly square mortises, and many woodworkers "drag" this along the bottom to achieve a relatively flat mortise bottom.  If you want to use only hand tools, a router plane would be capable of achieving this, coming from both directions to get into the corners.