I am seeing GeomNav1002 warnings when low energy neutrons generated in a thick lead radiator reach an azimuthally replicated volume. The issue occurs with geant4.10.7 (but likely before as well). I would like to create a minimal example but have been unable to reproduce it in any of the examples. If you have some hints as to where to debug, I will happily try to reproduce this.

Here is the core warning:

G4WT0 >    41 1.06e+03     75.2 2.47e+04  0.000259        0    0.598       698 donutLead_physical CoupledTransportation
G4WT0 >    42 1.04e+03     56.4 2.47e+04  0.000259        0     32.1       730 donutSystem_logical_PV CoupledTransportation
G4WT0 >    43      812     -111 2.47e+04  0.000259        0      286  1.02e+03 logicMother_PV CoupledTransportation
G4WT0 >    44     -130     -810 2.49e+04  0.000259        0 1.19e+03  2.21e+03 pionDetectorSystem_logical_PV CoupledTransportation
G4WT0 >    45     -593 -1.15e+03  2.5e+04  0.000259        0      584  2.79e+03 pionDetectorTSOpenSector_logical_PV CoupledTransportation
G4WT0 > 
-------- WWWW ------- G4Exception-START -------- WWWW -------
*** G4Exception : GeomNav1002
      issued by : G4PathFinder::ReportMove()
Endpoint moved between value returned by ComputeStep() and call to Locate(). 
          Change of  (End) Position / G4PathFinder::Locate is 9.999994268616646e-08 mm long
          and its vector is (7.921528322185623e-08,5.86951500736177e-08,-1.67274265550077e-08) mm 
          Endpoint of ComputeStep() was     (-592.8257463279759,-1152.573179067321,25000.00000001673)
          and current position to locate is (-592.8257462487607,-1152.573179008625,25000)
*** This is just a warning message. ***
-------- WWWW -------- G4Exception-END --------- WWWW -------

G4WT0 >    46     -593 -1.15e+03  2.5e+04  0.000259        0        0  2.79e+03 pionDetectorTSOpenSector_logical_PV nKiller
G4WT0 > 

Volume pionDetectorTSOpenSector_logical_PV is implemented in gdml as a replicavol (which as been checked for overlaps):

<solids>
  <tube name="pionDetectorTSOpenSector_solid"
    rmin="pionDetectorTSInnerRadius"
    rmax="pionDetectorTSOuterRadius"
       z="pionDetectorTSLength"
    startphi="-0.5*PHI/2.0" deltaphi="0.5*PHI" aunit="deg"/>
  <tube name="pionDetectorTSAllSectors_solid"
    rmin="pionDetectorTSInnerRadius"
    rmax="pionDetectorTSOuterRadius"
       z="pionDetectorTSLength"
    startphi="-PHI/2.0" deltaphi="360.0" aunit="deg"/>
</solids>

<structure>

  <volume name="pionDetectorTSOpenSector_logical">
    <materialref ref="G4_AIR"/>
    <solidref ref="pionDetectorTSOpenSector_solid"/>
    <physvol name="pionDetectorTS_physical">
      <file name="pion/TS/pionDetectorTS.gdml"/>
      <position name="pionDetectorTS_position" x="pionDetectorTSRadialPosition"/>
      <rotation name="pionDetectorTS_rotation" y="-90.0" z="90" unit="deg"/>
    </physvol>
    <auxiliary auxtype="Alpha" auxvalue="0.1"/>
  </volume>

  <volume name="pionDetectorTSAllSectors_logical">
    <materialref ref="G4_AIR"/>
    <solidref ref="pionDetectorTSAllSectors_solid"/>
    <!--loop for="i" from="0" to="1" step="1">
      <physvol name="pionDetectorTSOpenSector_physical[i]">
        <volumeref ref="pionDetectorTSOpenSector_logical"/>
        <rotation z="-PHI0-i*PHI" unit="deg"/>
      </physvol>
    </loop-->
    <replicavol number="2">
      <volumeref ref="pionDetectorTSOpenSector_logical"/>
      <replicate_along_axis>
        <direction phi="1"/>
        <width value="PHI" unit="deg"/>
        <offset value="PHI0-PHI/2.0" unit="deg"/>
      </replicate_along_axis>
    </replicavol>
    <auxiliary auxtype="Alpha" auxvalue="0.1"/>
  </volume>

</structure>

Other useful tidbits:
- switching from QGSP_BERT to QGSP_BERT_HP make the problem go away (but this could be because the neutrons cross sections are different)
- switching from replicavol to looped placement (see above) makes the problem go away
- switching off the parallel world does NOT solve the problem 
- making the replicated volumes slightly smaller to avoid overlapping faces does NOT solve the problem