deal with the surge in petitioners, plus cooperation between the federal and state gov'ts when it comes to placing and following those allowed into the country....not unannounced and unanticipated "Drop Offs" in the middle of the night.
As for limits on asylum, there are 'Presidential Limits' set...note this excerpt from the link above...
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Of the 62,500 admissions determined by the president for FY 2021, the regional allocations are shown in Table 3 below. Given the slow pace of admissions, it is unlikely that any of these allocations will be met.
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We aren't being "overrun"...it could be argued that we are "under maned" given the demographic issue I brought up and its implications for GDP growth.