A Study of ASP388B Given by Itself and With Standard Therapies in Participants With Solid Tumors

Inclusion Criteria: For the ASP388B monotherapy dose escalation (excluding the tumor-specific backfill participants), the following criteria apply: * Participant has a confirmed diagnosis of locally advanced unresectable or metastatic solid tumors. * Participant has progressed on, is ineligible for, or has refused all available standard therapies (no limit to the number of prior treatment regimens). * Prior exposure to B7-H3, STING agonist or TopI directed therapy is allowed. * Participant must have one of the following malignancies (for all tumor types, any compounds of neuroendocrine histology is ineligible): HNSCC, ESCC, mCRPC, sqNSCLC, SCLC * For mCRPC * Participants with histologically or cytologically confirmed mCRPC who are refractory to a novel antiandrogen therapy (including but not limited to abiraterone, enzalutamide and/or apalutamide) and have failed at least 1 (but not more than 2) taxane regimens (or who are deemed medically unsuitable to be treated with a taxane regimen or have actively refused treatment with a taxane regimen). * Participants must have undergone bilateral orchiectomy or must be on continuous ADT with a GnRH agonist or antagonist. * Total serum testosterone ≤ 50 ng/dL or 1.7 nmol/L * Evidence of progressive disease, defined as ≥ 1 PCWG3 criteria: PSA level ≥ 1 ng/mL that has increased on at least 2 successive occasions at least 1 week apart; Nodal or visceral progression as defined by RECIST v1.1 with PCWG3 modifications; Appearance of ≥ 2 new lesions in bone scan * Participants with mCRPC can be enrolled without measurable disease. * For sqNSCLC * Participants with known EGFR, ALK, ROS, BRAF or other actionable mutations are eligible if treated with mutation targeted therapy and have progressed, relapsed or discontinued treatment due to toxicity. * For SCLC * Metastatic or extensive stage (EC-SCLC) For the ASP388B 2L+HNSCC monotherapy dose expansion (including the tumor-specific backfill participants from the monotherapy dose escalation), the following criteria apply: * Participant has histologically or cytologically confirmed HNSCC. * Tumors arising from the nasopharynx are excluded. * Known PD-1/PD-L1 status and any HPV status is eligible. * Participant has evidence of radiographic progression on or after the last regimen received. * Participant has locally advanced or metastatic disease that is not amenable to curative intent treatment. * Participant has progressed, relapsed or discontinued treatment due to toxicity after local guidelines/SOC regimen for locally advanced or metastatic disease, and has received ≤ 3 prior lines of anticancer therapy in the locally advanced or metastatic setting. * Neoadjuvant or adjuvant cytotoxic regimens will count as a prior regimen if relapsed or progressed ≤ 6 months after completion. * No prior exposure to B7-H3, STING agonist or TopI directed therapy is allowed. For the ASP388B 2L+ESCC monotherapy dose expansion (including the tumor-specific backfill participants from the monotherapy dose escalation), the following criteria apply: * Participant has histologically or cytologically confirmed ESCC. * Known PD-1/PD-L1 status and any HER2 status is eligible. * Participant has evidence of radiographic progression on or after the last regimen received. * Participant has locally advanced or metastatic disease that is not amenable to curative intent treatment. * Participant has progressed, relapsed or discontinued treatment due to toxicity after local guidelines/SOC regimen for locally advanced or metastatic disease, and has received ≤ 3 prior lines of anticancer therapy in the locally advanced or metastatic setting. * Neoadjuvant or adjuvant cytotoxic regimens will count as a prior regimen if relapsed or progressed ≤ 6 months after completion. * No prior exposure to B7-H3, STING agonist or TopI directed therapy is allowed. For the ASP388B 1L HNSCC (Excluding NPC and Salivary Gland Tumors) combination therapy dose escalation and expansion (including the tumor-specific backfill participants from combination therapy dose escalation), the following criteria apply: (ASP388B + Pembrolizumab + Carboplatin + 5-FU): * Participant has histologically or cytologically confirmed HNSCC excluding nasopharyngeal cancer (NPC) and salivary gland tumors. * Known PD-1/PD-L1 status and any HPV status is eligible * Participant has recurrent or metastatic disease that is incurable by local therapies. * Participant has had no prior systemic therapy administered with the exception of systemic therapy completed \> 6 months prior if given as part of multimodal treatment with curative intent for nonmetastatic disease stages. * No prior exposure to B7-H3, STING agonist or TopI directed therapy is allowed. For the ASP388B 1L ESCC combination therapy dose escalation and expansion (including the tumor-specific backfill participants from combination therapy dose escalation), the following criteria apply: (ASP388B + Pembrolizumab + Oxaliplatin + 5-FU): * Participant has histologically or cytologically confirmed ESCC. * Known PD-1/PD-L1 status and any HER2 status is eligible * Participant has had no prior systemic therapy administered with the exception of systemic therapy completed \> 6 months prior if given as part of multimodal treatment with curative intent for nonmetastatic disease stages. * No prior exposure to B7-H3, STING agonist or TopI directed therapy is allowed. For the ASP388B 1L ESCC combination therapy dose escalation and expansion * Participant has a predicted life expectancy ≥ 12 weeks. * Participant has at least 1 measurable lesion per RECIST v1.1. Lesions situated in a previously irradiated area are considered measurable if progression has been demonstrated in such lesions. * Participant has an ECOG performance status of 0 or 1. * Participant has adequate organ function as indicated by laboratory values (If a participant has received a recent blood transfusion, the laboratory tests must be