Inclusion Criteria:

          -  For Dose Escalation: Patients must have histologically or cytologically confirmed
             advanced or metastatic gastrointestinal (GI) cancers with Response Evaluation Criteria
             in Solid Tumors 1.1 (RECIST1.1) measurable disease who have progressed on at least one
             prior treatment for metastatic disease and for whom FOLFIRI is considered a reasonable
             treatment option. Patients with mismatch repair deficiency should have progressed on
             immunotherapy

          -  For Dose Expansion: Patients must have either:

               -  Colorectal cancer who have previously progressed on irinotecan and tolerated an
                  irinotecan dose equal to or greater than the recommended phase 2 dose (RP2D). If
                  they have mismatch repair deficiency they should have progressed on immunotherapy
                  OR

               -  Gastroesophageal cancer who have progressed on at least one first-line therapy
                  for metastatic disease. If they have mismatch repair deficiency they should have
                  progressed on immunotherapy

          -  For Dose Expansion: Patients be willing to undergo biopsies for research purposes
             only. The accessible tumor can be the primary or metastatic tumor site. Both research
             biopsies should be taken from the same tumor site

          -  Patients must have progressive disease on at least first-line therapy for metastatic
             disease. Previous treatment with irinotecan is allowed

          -  Eastern Cooperative Oncology Group (ECOG) performance status =< 1

          -  Leukocytes >= 3,000/mcL

          -  Absolute neutrophil count >= 1,500/mcL

          -  Platelets >= 100,000/mcL

          -  Total bilirubin =< 1.5 x institutional upper limit of normal (ULN)

          -  Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase
             [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase [SGPT])
             =< 3 x institutional ULN

          -  Glomerular filtration rate (GFR) >= 60 mL/min/1.73 m^2 using the Chronic Kidney
             Disease Epidemiology Collaboration (CKD-EPI) equation

          -  Human immunodeficiency virus (HIV)-infected patients on effective anti-retroviral
             therapy, that does not interact with study therapy, with undetectable viral load
             within 6 months are eligible for this trial

          -  For patients with evidence of chronic hepatitis B virus (HBV) infection, the HBV viral
             load must be undetectable on suppressive therapy that does not interact with study
             therapy, if indicated

          -  Patients with a history of hepatitis C virus (HCV) infection must have been treated
             and cured. For patients with HCV infection who are currently on treatment, they are
             eligible if they have an undetectable HCV viral load and HCV therapy does not interact
             with study therapy

          -  Patients with treated brain metastases are eligible if follow-up brain imaging after
             central nervous system (CNS)-directed therapy shows no evidence of progression

          -  Patients with a prior or concurrent malignancy whose natural history or treatment does
             not have the potential to interfere with the safety or efficacy assessment of the
             investigational regimen are eligible for this trial

          -  Patients with known history or current symptoms of cardiac disease, or history of
             treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac
             function using the New York Heart Association Functional Classification. To be
             eligible for this trial, patients should be class 2B or better

          -  The effects of BAY 1895344 on the developing human fetus are unknown. For this reason
             and because DNA-damage response inhibitors agents as well as other therapeutic agents
             used in this trial, 5-FU and irinotecan, are known to be teratogenic, women of
             child-bearing potential and men must agree to use adequate contraception (hormonal or
             barrier method of birth control; abstinence) prior to study entry and for the duration
             of study participation and for 6 months after completion of BAY 1895344 administration

               -  Should a woman become pregnant or suspect she is pregnant while she or her
                  partner is participating in this study, she should inform her treating physician
                  immediately. Men treated or enrolled on this protocol must also agree to use
                  adequate contraception prior to the study, for the duration of study
                  participation, and 6 months after completion of study treatment administration

          -  Ability to understand and the willingness to sign a written informed consent document.
             Participants with impaired decision-making capacity (IDMC) who have a
             legally-authorized representative (LAR) and/or family member available will also be
             eligible

        Exclusion Criteria:

          -  Patients with a history of prior treatment with an ATR inhibitor

          -  Patients with a history of other malignancy that could affect compliance with the
             protocol or interpretation of the results

          -  Patients who have had chemotherapy or radiotherapy within 3 weeks (6 weeks for
             nitrosoureas or mitomycin C) prior to entering the study

          -  Patients who have not recovered from adverse events due to prior anti-cancer therapy
             (i.e., have residual toxicities > grade 1) with the exception of alopecia

          -  Patients who are receiving any other investigational agents

          -  History of hypersensitivity or allergic reactions attributed to compounds of similar
             chemical or biologic composition to BAY 1895344, 5-FU, leucovorin, or irinotecan

          -  Patients receiving any medications that are substrates of CYP3A4 with a narrow
             therapeutic window or strong inhibitors/inducers of CYP3A4 are ineligible, if they
             cannot be transferred to alternative medication. Because the lists of these agents are
             constantly changing, it is important to regularly consult a frequently-updated medical
             reference. As part of the enrollment/informed consent procedures, the patient will be
             counseled on the risk of interactions with other agents, and what to do if new
             medications need to be prescribed or if the patient is considering a new
             over-the-counter medicine or herbal product

          -  Patients with uncontrolled intercurrent illness

          -  Patients with psychiatric illness/social situations that would limit compliance with
             study requirements

          -  Gastrointestinal pathology or history that adversely impact the ability to take or
             absorb oral medication

          -  Pregnant women are excluded from this study because BAY 1895344 as a DNA-damage
             response inhibitor may have the potential for teratogenic or abortifacient effects.
             Because there is an unknown but potential risk for adverse events in nursing infants
             secondary to treatment of the mother with BAY 1895344 breastfeeding should be
             discontinued if the mother is treated with BAY 1895344 and for 4 months after end of
             treatment. These potential risks may also apply to other agents used in this study

          -  Patients who were unable to tolerate prior irinotecan treatment are excluded from this
             study

          -  Patients with a corrected QT (QTc) interval >= 470 msec are excluded from this study